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HACETTEPE JOURNAL OF

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A Bimonthly PublicationVolume 42 Issue 1

2013

ISSN 1303 5010

HACETTEPE JOURNAL OF

MATHEMATICS AND

STATISTICS

Volume 42 Issue 1

February 2013

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Hacettepe Journal of Mathematics and Statistics

Cilt 42 Sayı 1 (2013)

ISSN 1303 – 5010

KUNYE

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HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS

YIL : 2013 SAYI : 42 - 1 AY : Subat

YAYIN SAHIBININ ADI : H. U. Fen Fakultesi Dekanlıgı adına

Prof. Dr. Kadir Pekmez

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Hacettepe Journal of Mathematics and Statistics

A Bimonthly Publication – Volume 42 Issue 1 (2013)

ISSN 1303 – 5010

EDITORIAL BOARD

EDITOR : Yucel TırasASSOCIATE EDITOR : Lawrence M. Brown

MEMBERS

Gary F. Birkenmeier (University of Louisiana at Lafayette, USA.)

G. C. L. Brummer (University of Cape Town, South Africa)

Cem Kadılar (Hacettepe University, Ankara, Turkey)

Varga Kalantarov (Koc University, Istanbul, Turkey)

Vladimir Levchuk (Siberian Federal University, Russia)

Cihan Orhan (Ankara University, Ankara, Turkey)

Ivan Reilly (University of Auckland, New Zealand)

Patrick Smith (Glasgow University, United Kingdom)

Bulent Sarac (Hacettepe University, Ankara, Turkey)

Alexander P. Sostak (University of Latvia, Riga, Latvia)

Agacık Zafer (Middle East Technical University, Ankara, Turkey)

Published by Hacettepe UniversityFaculty of Science

CONTENTS

Mathematics

E. Kılıc and P. Stanica

General Approach in Computing Sums of Products of Binary Sequences . . . . . . . 1

S. Ebru Das.

Dynamics of a Nonlinear Rational Difference Equation . . . . . . . . . . . . . . . . . . . . . . . . 9

Suleyman Solak and Mustafa Bahsi

On the Spectral Norms of Toeplitz Matrices with Fibonacci

and Lucas Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Mujahid Abbas, Talat Nazır and B. E. Rhoades

Fixed Points of Multivalued Mapping Satisfying Ciric Type

Contractive Conditions in G-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Hasan Atik

Cofibration Category and Homotopies of Three–Crossed Complexes . . . . . . . . . . . 31

A. Dilek Gungor Maden, Ivan Gutman and A. Sinan Cevik

Bounds for Resistance–Distance Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

Xianping Liu, Dajing Xiang, K. P. Shum and Jianming Zhan

Soft Rings Related to Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Derya Keskin Tutuncu and Sultan Eylem Toksoy

Absolute Co-Supplement and Absolute Co-Coclosed Modules . . . . . . . . . . . . . . . . . . 67

Yiqiu Du and Yu Wang

A Result on Generalized Derivations in Prime Rings . . . . . . . . . . . . . . . . . . . . . . . . 81

Statistics

Ahmet Kara

Dynamics of Performance in Higher Education an AppliedStochastic Model and a Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87

MATHEMATICS

Hacettepe Journal of Mathematics and StatisticsVolume 42 (1) (2013), 1 – 7

GENERAL APPROACH IN COMPUTING

SUMS OF PRODUCTS OF BINARY

SEQUENCES

E. Kılıc∗ , P. Stanica†

Received 10 : 02 : 2010 : Accepted 16 : 12 : 2011

Abstract

In this paper we find a general approach to find closed forms of sumsof products of arbitrary sequences satisfying the same recurrence withdifferent initial conditions. We apply successfully our technique to sumsof products of such sequences with indices in (arbitrary) arithmeticprogressions. It generalizes many results from literature. We proposealso an extension where the sequences satisfy different recurrences.

Keywords: Second order recurrences; Sums; Products

2000 AMS Classification: 11B37, 11C20

1. Introduction

We consider a generic (nondegenerate, that is, δ =√p2 − 4q 6= 0) binary recurrence

satisfying

(1.1) Xn+1 = pXn − qXn−1, n ∈ Z

with some initial conditions. Let α, β be the roots of the equation x2 − px+ q = 0, andso, α+ β = p, αβ = q, δ = α− β. We associate the companion Lucas sequence Ln whichalso satisfies (1.1) together with L0 = 2, L1 = p, and so Ln = αn + βn.

Let {U (j)n }pj=1 be a set of p binary sequences, all of which will satisfy the recurrence

(1.1) with some initial conditions, such that the Binet formula for these sequences is

U (j)n = Ajα

n +Bjβn, n ∈ Z,

where Aj =U

(j)1 −U(j)

0 β

δ, Bj =

U(j)0 α−U(j)

1δ

.

∗TOBB Economics and Technology University, MathematicsDepartment, 06560 Sogutozu,Ankara, Turkey.

E-mail: [email protected]†Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, USA.

E-mail: [email protected]

2 E. Kilic, P. Stanica

For easy notation, we will denote the recurrence {Xn} given by (1.1) by {Xn (p, q, a, b)}where a = X0 and b = X1 are initial conditions of it.

Several authors investigated products of two terms of a sequence or products of twosequences, and also, the sums of these products. As a first example, note that the sumof square terms of Fibonacci numbers [7, 8, 12] is

n∑i=1

F 2i = FnFn+1.

The sum of products of variable subscripted terms of certain second order recurrenceshave been considered by several authors. For example (see [11])

n∑i=1

FiFi+2 =F2n+1F2n+2 − 1,

n∑i=1

FiFi+1 =F 22n+1 − 1,

n∑i=1

F2i−1F2i+3 =(3F 2

2n+2 − 2F 22n+1 + 7n− 1

)/5.

Certainly, the classical Fibonacci, Fn and Pell numbers Pn are Fn = Xn (1,−1, 0, 1)and Pn = Xn (2,−1, 0, 1). Generalizations of the above sums by taking different recur-rences and their variable subscripted terms have also been studied. For example, in [9],the author found

∑ni=1 FiPi. Melham [10] looked at the sum of the squares of the se-

quence {Xn (2, 1, 0, 1)}. Recently, in [2, 3, 4, 5, 6], the authors gave several formulas forsums of squares of even and odd Fibonacci, Lucas and Pell-Lucas numbers, and theirsums of products of even and odd subscripted terms. Also the authors of [1] establishedseveral formulas for sums and alternating sums of products of certain subscripted termsof recurrences {Xn (p, q, 0, 1)} and {Xn (p, q, 2, p)} .

It is our goal in this paper to propose a general approach for the theory of closed formsfor sums of products of nondegenerate second-order recurrent sequences, thus generalizingmany of these kind of results that the reader can find scattered throughout the literature.

2. Main Results

Let P(n) be the power set of {1, 2, . . . , n}, that is the set of all subsets of {1, 2, . . . , n}.Given a sequence of p functions fj(i), j = 1, . . . , p, for all M ∈ P(p), we let FM (i) =∑`∈M f`(i), F∅(i) = 0, and for simplicity, F (i) = F{1,...,p}(i) =

∑p`=1 f`(i). Let us define

a set of twisted product sequences, indexed by the sets M ∈ P(p), in the following way:

for a set M ∈ P(p), we let W(M)n be a (M-twisted product) rational sequence satisfying

(1.1) with the Binet formula

W (M)n =

∏j∈M

Aj∏k 6∈M

Bk

αn +

∏j∈M

Bj∏k 6∈M

Ak

βn.

Further, we use M = {1, 2, . . . , p} \M , for the complement of the set M in {1, 2, . . . , p}.We shall first show that W

(M)n is a rational sequence, even more precise that W

(M)n ∈

1

δ2p−1 Z.

2.1. Lemma. For any integer n, the twisted product sequences satisfy

W (M)n = qnW

(M)−n .

Proof. Straightforward using the Binet formula. �

General Approach in Computing Sums of Products of Binary Sequences 3

2.2. Theorem. For p, n ∈ Z, we have

W (M)n ∈ 1

δ2p−1 Z.

Proof. We will prove the claim by induction. First, we let p = 2, and consider twosequences Un = A1α

n + B1βn, Vn = A2α

n + B2βn (for simplicity of notations). We

write the superscript sets as {a, . . .} instead of ({a, . . .}). The associated twisted productsequences are

W {1,2}n = A1A2αn +B1B2β

n,

W {1}n = A1B2αn +B1A2β

n,

W {2}n = A2B1αn +A1B2β

n,

W ∅n = B1B2αn +A1A2β

n.

Since our index n runs through the entire set of integers, by Lemma 2.1, it will be

sufficient to consider only the case of W{1,2}n , and W

{1}n .

First, using the expressions for A1, A2, B1, B2 in terms of initial conditions of Un, Vn,and simplifying, we get

δ2(A1A2 +B1B2) = 2U1V1 − p(U0V1 + U1V0) + (p2 − 2q)U0V0 ∈ Z

δ2(A1A2α+B1B2β) = pU1V1 − 2q(U1V0 + U0V1) + pqU0V0 ∈ Z.

Further,

δ2(A1B2 +B1A2) = p(U1V0 + V1U0)− 2qU0V0 − 2U1V1 ∈ Z

δ2(A1B2α+B1A2β) = (p2 − 2q)U1V0 − p(U0V0q + U1V1) + 2U0V1q ∈ Z.

Now, let Un ∈ Z and, from the induction step, assume that Vn ∈ 1

δ2p−1 Z. As before,

writing δ2W{1,2}0 , δ2W

{1,2}1 , δ2W

{1}0 , δ2W

{1}1 in terms of U0, U1, V0, V1, we see that each

term in these expressions contains only one factor based on either V0, or V1 ∈ 1

δ2p−1 Z,

and therefore W{1,2}i ,W

{1}i ∈ 1

δ2p Z, i = 0, 1. Certainly, since the initial terms of the

twisted product sequences are in 1δ2p Z, so is W

(M)n . �

We show now our general approach to finding sums of products of recurrences.

2.3. Theorem. Given a set of p functions fj(i), j = 1, . . . , p, such that fj(i)− f`(i) isa function of j, ` only and it does not depend on i, we have

n∑i=0

p∏j=1

U(j)

fj(i)=

1

2

∑M∈P(p)

n∑i=0

qF (i)−FM (i)W(M)

2FM (i)−F (i).

Proof. First, we associate to every set M ∈ P(p) a bit string ε of length p in the usualmanner (a 1 bit appears in the bit string if and only if its corresponding position appearsin M , otherwise the bit is 0). For ε ∈ Zp2, we let wt(ε) to be the Hamming weight ofthe bit string ε, that is, the number of 1’s in its expression, and supp(ε) = {i1 < i2 <. . . < iwt(ε)} to be the support of ε (the positions where 1’s appear in ε). Certainly,supp(ε) ⊆ {1, 2, . . . , p}.


Next, we compute the product

p∏j=1

U(j)

fj(i)=

p∏j=1

(Ajα

fj(i) +Bjβfj(i)

)=

∑ε=(ε1,...,εp)∈Zp2

p∏j=1

Aεjj B

1−εjj αεjfj(i)β(1−εj)fj(i)

=1

2

∑ε∈Zp2

∏j∈supp(ε)

∏k 6∈supp(ε)

AjBkα∑j∈supp(ε) fj(i)β

∑k 6∈supp(ε) fk(i)

+∏

j∈supp(ε)

∏k 6∈supp(ε)

AkBjβ∑j∈supp(ε) fj(i)α

∑k 6∈supp(ε) fk(i)

=

1

2

∑M∈P(p)

(αβ)∑j 6∈M fj(i)

∏j∈M

∏k 6∈M

AjBkα∑j∈M fj(i)−

∑j 6∈M fj(i)

+∏j∈M

∏k 6∈M

AkBjβ∑j∈M fj(i)−

∑j 6∈M fj(i)

=

1

2

∑M∈P(p)

qF (i)−FM (i)W(M)

2FM (i)−F (i),

from which our theorem follows easily. �

Obviously, if the sum∑ni=0 q

F (i)−FM (i)W(M)

2FM (i)−F (i) can be simplified, then the pre-

vious theorem takes quite an attractive form. The rest of the paper is devoted in findingvarious functions fj for which such a sum can be computed. Many papers are investi-gating sums of products of very few recurrences (mostly two) where the indices are veryspecific linear functions. We will attack this case in its full generality here and solve itcompletely, by taking fj to be arbitrary linear functions.

Let Wn be our generic sequence satisfying (1.1) such that Wn = Aαn + Bβn, andrecall that Ln = αn + βn is the companion Lucas sequence.

2.4. Lemma. For a, b, c, d ∈ Z, we have the generating function

n∑i=0

xa+biWc+di = xaqdxb(n+2)Wc+dn − xb(n+1)Wc+d(n+1) − xbqdWc−d +Wc

x2bqd − xbLd + 1.


Proof. Using Binet formula for Wn, we obtain

n∑i=0

xbiWc+di = Aαcn∑i=0

(xbαd)i +Bβcn∑i=0

(xbβd)i

= Aαc(xbαd)n+1 − 1

xbαd − 1+Bβc

(xbβd)n+1 − 1

xbβd − 1

=Axb(n+2)βdαc+d(n+1) −Axb(n+1)αc+d(n+1) −Axbαcβd +Aαc

+Bxb(n+2)αdβc+d(n+1) −Bxb(n+1)βc+d(n+1) −Bxbβcαd +Bβc

x2b(αβ)d − xb(αd + βd) + 1

=qdxb(n+2)(Aαc+dn +Bβc+dn)− xb(n+1)(Aαc+d(n+1) +Bβc+d(n+1))

−qdxb(Aαc−d +Bβc−d) + (Aαc +Bβc)

x2bqd − xbLd + 1

=qdxb(n+2)Wc+dn − xb(n+1)Wc+d(n+1) − xbqdWc−d +Wc

x2bqd − xbLd + 1.

�

Taking Wn = un = Xn (p, q, 0, 1) , we reach at the following result:

n∑i=0

(−1)i ur+4i = (−1)nυ4n+r+2 + ur−2

v2

where vn = Xn (p, q, 2, p) . One can also find this result in [1, Lemma 5].Let fj(i) = aj + bji be linear functions. Under these conditions,

F (i)− FM (i) =

p∑j=1

(aj + bji)−∑j∈M

(aj + bji)

=

∑j 6∈M

aj

+

∑j 6∈M

bj

i = a(M) + b(M)i,

where we use the notations a(M) =∑j 6∈M aj and b(M) =

∑j 6∈M bj . We shall also use

a(M) =∑j∈M aj , b

(M) =∑j∈M bj . Further,

2FM (i)− F (i) =∑j∈M

(aj + bji)−∑j 6∈M

(aj + bji)

=(a(M) − a(M)

)+(b(M) − b(M)

)i.

Applying Lemma 2.4 with x := q, a := a(M), b := b(M), c := a(M) − a(M), d :=

b(M) − b(M), and using Theorem 2.3 we obtain our next result.


2.5. Theorem. Given a set of linear functions fj(i) = aj + bji, and binary sequences

U(j)k satisfying (1.1) with some initial conditions, we have

n∑i=0

p∏j=1

U(j)

fj(i)=

1

2

∑M∈P(p)

qa(M)

qb(M)+b(M)(n+1)W

(M)

a(M)−a(M)+n(b(M)−b(M))

−qb(M)(n+1)W

(M)

a(M)−a(M)+(n+1)(b(M)−b(M))

−qb(M)

W(M)

a(M)+a(M)−b(M)−b(M)+W

(M)

a(M)−a(M)

qb(M) − qb(M)Lb(M)−b(M) + 1.

3. A Particular Case

To understand our general result better, we shall consider now a particular case of twobinary recurrences, which is the case most often encountered in literature. Let Un, Vn betwo binary recurrent sequences satisfying (1.1) with some initial conditions. The Binetformula indicates that

Un = A1αn +B1β

n,

Vn = A2αn +B2β

n,

where A1 = U0β−U1β−α , B1 = U1−U0α

β−α , A2 = V0β−V1β−α , B2 = V1−V0α

β−α .

As before, we take the twisted productsW{1,2}n ,W

{1}n , satisfying (1.1), with initial con-

ditionsW{1,2}0 = A1A2+B1B2,W

{1,2}1 = A1A2α+B1B2β, W

{1}0 = A1B2+B1A2,W

{1}1 =

A1B2α+B1A2β, so that W{1,2}n = A1A2α

n +B1B2βn, and W

{1}n = A1B2α

n +B1A2βn.

From Theorem 2.2 we know that W{1}n ,W

{1,2}n ∈ 1

δ2Z. We next consider the example

f1(i) = r + ki, f2(i) = s+ ki.

3.1. Theorem. Let k, r, s be fixed integers. We haven∑i=0

Ur+kiVs+ki = qsW{1}r−s

qk(n+1) − 1

qk − 1

+q2kW

{1,2}r+s+2kn −W

{1,2}r+s+2k(n+1) − q

2kW{1,2}r+s−2k +W

{1,2}r+s

q2k − L2k + 1.

Proof. First,

Ur+kiVs+ki = (A1αr+ki +B1β

r+ki)(A2αs+ki +B2β

s+ki)

= (A1A2αr+s+2ki +B1B2β

r+s+2ki)

+ (A1B2αr+kiβs+ki +A2B1α

s+kiβr+ki)

= W{1,2}r+s+2ki + qs+ki(A1B2α

r−s +A2B1βr−s)

= W{1,2}r+s+2ki + qs+kiW

{1}r−s.(3.1)

In the notations of Theorem 2.5, the previous product will be

1

2

(qf1(i)+f2(i)W ∅−F (i) + qf2(i)W

{1}f1(i)−f2(i) + qf1(i)W

{2}f2(i)−f1(i)

+W{1,2}f1(i)+f2(i)

)= W

{1,2}f1(i) + qf2(i)W

{1}f1(i)−f2(i).


Using (3.1), we separate the sum∑ni=0 Ur+kiVs+ki into two sums. First,

(3.2)

n∑i=0

qs+kiW{1}r−s = qsW

{1}r−s

n∑i=0

(qk)i = qsW{1}r−s

qk(n+1) − 1

qk − 1,

(we could have also used Lemma 2.4 with x := q, a = s, b = k and c = r − s, d = 0).Next, using Lemma 2.4 with x := q, a = b = 0 and c = r + s, d = 2k, we get

n∑i=0

Wt+`i =q`Wt+`n −Wt+`(n+1) − q`Wt−` +Wt

q` − L` + 1.

and the second sum becomesn∑i=0

W{1,2}r+s+2ki =

q2kW{1,2}r+s+2kn −W

{1,2}r+s+2k(n+1) − q

2kW{1,2}r+s−2k +W

{1,2}r+s

q2k − L2k + 1,

which finishes the proof of our theorem. �

If we take un = Xn (p, q, 0, 1) and vn = Xn (p, q, 2, p) (p 6= 0,√p2 − 4q 6= 0), then by

required arrangements, we obtain for k = 2n∑i=0

ur+2ivs+2i =v4n+r+s+2 − vr+s−2 − p (n+ 1) qrus−r

p (p2 − 4q)

which is the main result of [1, Theorem 1].

References

[1] Belbachir H. and Bencherif, F. Sums of products of generalized Fibonacci and Lucas num-

bers, Arxiv:0708.2347v1, 2009.[2] Cerin, Z. Sums of products of generalized Fibonacci and Lucas numbers, Demons. Math.,

42 (2), 247–258, 2009.

[3] Cerin, Z. On Sums of Products of Horadam Numbers, Kyungpook Math. J., 49, 483–492,2009.

[4] Cerin, Z. and Gianella G.M. On sums of squares of Pell-Lucas numbers, Integers 6A15, 16pp., 2006.

[5] Cerin, Z. Alternating sums of Fibonacci products, Atti Semin. Mat. Fis. Univ. Modena

Reggio Emilia 53 (2), 331–344, 2005.[6] Cerin, Z. Some alternating sums of Lucas numbers, Cent. Eur. J. Math. 3 (1) , 1–13, 2005.

[7] Kılıc, E. Sums of the squares of terms of sequence {un}, Proc. Indian Acad. Sci. 118 (1),

27–41, 2008.[8] Koshy, T. Fibonacci and Lucas numbers with applications, Pure and Appl. Math., Wiley-

Interscience, New York, 2001.

[9] Mead, D.G. Problem B-67, Fibonacci Quart. 3 (4), 326–327, 1965.[10] Melham, R.S. On sums of powers of terms in a linear recurrence, Portugal. Math. 56 (4),

501–508, 1999.

[11] Rao, K.S. Some properties of Fibonacci numbers, The Amer. Math. Monthly, 60, 680–684,1953.

[12] Vajda, S. Fibonacci & Lucas numbers, and the golden section, John Wiley & Sons, Inc.,New York, 1989.


DYNAMICS OF A NONLINEAR RATIONAL

DIFFERENCE EQUATION

S. Ebru Das.∗


Abstract

In this paper, we investigate the dynamical properties of the followingnonlinear difference equation:

xn+1 =xanxn−2xn−3 + xnxn−2x

an−3 + 1

xanxn−3 + xnxa

n−3 + 1, n = 0, 1, . . .

Keywords: Difference equation, Stability, Global Stability, Oscillation.

2000 AMS Classification: 39A10.

1. Introduction

Recently, there has been a great interest in studying the qualitative behavior of rationaldifference equations. Berenhaut et al.[4] has showed that the unique positive equilibriumy = 1 of the difference equation:

yn =yn−k + yn−m

1 + yn−kyn−m, n = 0, 1, . . .

is globally asymptotically stable.

Chen et al.[5] investigated the dynamical properties of the following fourth-order nonlin-ear difference equation:

xn+1 =xan−2 + xn−3

xan−2xn−3 + 1

, n = 0, 1, . . .

with nonnegative initial conditions and a ∈ [0, 1).

Das [6] investigate the qualitative behavior of the following fourth-order difference equa-tion:

xn+1 =xn−1x

an−2 + xn−1x

an−3 + 1

xan−2 + xa

n−3 + 1, n = 0, 1, . . .

∗Yıldız Technical University, Department of Mathematics, Istanbul, Turkey.E-mail: [email protected]

10 S. E. Das.

where a ∈ (0,∞) and the initial conditions x−3, x−2, x−1, x0 ∈ (0,∞). For more work,see [1, 2, 3, 7, 8, 9, 10].

To be motivated by the above studies, in this paper, we consider the following nonlineardifference equation:

(1.1) xn+1 =xanxn−2xn−3 + xnxn−2x

an−3 + 1

xanxn−3 + xnxa

n−3 + 1, n = 0, 1, . . .

where a ∈ (0,∞) and the initial conditions are arbitrary positive real numbers. It is easyto see that the positive equilibrium x = 1 of Eq.(1.1) satisfies x = (2xa+2+1)/(2xa+1+1).

In the following, we state some main definitions used in this paper.

1.1. Definition. A positive semi-cycle of a solution {xn}∞n=−3 of Eq.(1.1) consists of a”string” of terms {x`, x`+1, ..., xm} all greater than or equal to the equilibrium x,

with ` ≥ −3 and m <∞ such that

either ` = −3 or ` > −3 and x`−1 < x

and

either m =∞ orm <∞ and xm+1 < x

A negative semi-cycle of a solution {xn}∞n=−3 of Eq.(1.1) consists of a ”string” of terms{x`, x`+1, ..., xm} all less than x,

with ` ≥ −3 and m <∞ such that

either ` = −3 or ` > −3 and x`−1 ≥ x

and

either m =∞ or m <∞ and xm+1 ≥ x

The length of a semi-cycle is the number of the total terms contained in it.

1.2. Definition. A solution {xn}∞n=−3 of Eq.(1.1) is said to be eventually trivial if xn

is eventually equal to x = 1 ; Otherwise the solution is said to be nontrivial. A solu-tion {xn}∞n=−3 of Eq.(1.1) is said to be eventually positive (negative) if xn is eventuallygreater (less) than x = 1.

2. Three Lemmas

Before to draw a qualitatively clear picture for the positive solutions of Eq.(1.1), wefirst establish three basic lemmas which will play a key role in the proof of our mainresults.

2.1. Lemma. A positive solution {xn}∞n=−3 of Eq.(1.1) is eventually equal to 1 if andonly if

(2.1) (x−2 − 1)(x−1 − 1)(x0 − 1) = 0

Proof. Assume that (2.1) holds. Then according to Eq.(??), it is easy to see that thefollowing conclusions hold:

(i) if x−2 = 1, then xn = 1 for n ≥ 40(ii) if x−1 = 1, then xn = 1 for n ≥ 40(ii) if x0 = 1, then xn = 1 for n ≥ 40

Dynamics of a Nonlinear Rational Difference Equation 11

Conversely, assume that

(2.2) (x−2 − 1)(x−1 − 1)(x0 − 1) 6= 0

Then one can show that

(2.3) xn 6= 1 for any n ≥ 1

Assume the contrary that for some N ≥ 1,

(2.4) xN = 1 and that xn 6= 1 for − 2 ≤ n ≤ N − 1

It is easy to see that

(2.5) 1 = xN =xaN−1xN−3xN−4 + xN−1xN−3x

aN−4 + 1

xaN−1xN−4 + xN−1xa

N−4 + 1

which implies (xaN−1xN−4 + xN−1x

aN−4)(xN−3 − 1) = 0. Obviously, this contradicts

(2.3). �

2.2. Remark. If the initial conditions do not satisfy Eq.(1.1), then, for any solution xn

of Eq.(1.1), xn 6= 1 for n ≥ −3. Here, the solution is a nontrivial one.

2.3. Lemma. Let {xn}∞n=−3 be a nontrivial positive solution of Eq.(1.1). Then thefollowing conclusions are true for n ≥ 0:

(a) (xn+1 − 1)(xn−2 − 1) > 0(b) (xn+1 − xn−2)(xn−2 − 1) < 0

Proof. It follows in light of Eq.(1.1) that

xn+1 − 1 =(xa

nxn−3 + xnxan−3)(xn−2 − 1)

xanxn−3 + xnxa

n−3 + 1, n = 0, 1, . . .

xn+1 − xn−2 =(1− xn−2)

xanxn−3 + xnxa

n−3 + 1, n = 0, 1, . . .

from which inequalities (a) and (b) follow. �

2.4. Lemma.

(i) If x−2, x−1, x0 > 1, then {xn}∞n=−3 has a positive semi-cycle with an infinitenumber of terms and it monotonically tends to the positive equilibrium pointx = 1.

(ii) If x−2, x−1, x0 < 1, then {xn}∞n=−3 has a negative semi-cycle with an infinitenumber of terms and it monotonically tends to the positive equilibrium pointx = 1.

Proof. (i) If x−2, x−1, x0 > 1, from Lemma 2.3.(a) and (b), for n ≥ −3

1 < x3k−2 < ... < x4 < x1 < x−2

1 < x3k−1 < ... < x5 < x2 < x−1

1 < x3k < ... < x6 < x3 < x0, k = 0, 1, . . .

Clearly, {xn}∞n=−3 has a positive semi-cycle with an infinite number of terms and mono-tonically decreasing for n ≥ 0. So the limit

(2.6) limn→∞

xn = L

exists and finite. Taking the limits on both sides of Eq.(1.1), we have

L =2La+2 + 1

2La+1 + 1

12 S. E. Das.

we can easily see that {xn}∞n=−3 tends to the positive equilibrium point x = 1.

(ii) If x−2, x−1, x0 < 1, from Lemma 2.3.(a) and (b), for n ≥ −2

x−2 < x1 < x4 < ... < x3k−2 < 1

x−1 < x2 < x5 < ... < x3k−1 < 1

x0 < x3 < x6 < ... < x3k < 1 k = 0, 1, . . .

Therefore, {xn}∞n=−3 has a negative semi-cycle with an infinite number of terms andmonotonically increasing for n ≥ 0. So the limit

(2.7) limn→∞

xn = M

exists and finite. Taking the limits on both sides of Eq.(1.1), we have

M =2Ma+2 + 1

2Ma+1 + 1

So, {xn}∞n=−3 tends to the positive equilibrium point x = 1. �

3. Main Results and their proofs

First we analyze the structure of the semi-cycles of nontrivial solutions of Eq.(1.1).Here we confine us to consider the situation of the strictly oscillatory solution of Eq.(1.1).

3.1. Theorem. Let {xn}∞n=−3 be a strictly oscillatory solution of Eq.(1.1). Then therule for the lengths of positive and negative semi-cycles of this solution to successivelyoccur is . . . 2+, 1−, 2+, 1−, 2+, 1−, . . . or . . . 2−, 1+, 2−, 1+, 2−, 1+, . . ..

Proof. By Lemma 2.3.(a) and (b), one can see that the length of a positive semi-cycleis not larger than 2 and the length of a negative semi-cycle is at most 2. Based on thestrictly oscillatory character of the solution, we see, for some p ≥ 0, that one of thefollowing two cases must occur:

Case1. xp−2 > 1, xp−1 < 1 and xp > 1

Case2. xp−2 > 1, xp−1 < 1 and xp < 1

If Case 1. Occurs, it follows from Lemma 2.3.(a) that

xp+1 > 1, xp+2 < 1, xp+3 > 1, xp+4 > 1, xp+5 < 1, xp+6 > 1, xp+7 > 1, xp+8 < 1, . . .

It means that the rule of the lengths of positive and negative semi-cycles of the solutionof Eq.(1.1) to occur successively is . . . 2+, 1−, 2+, 1−, 2+, 1−, . . ..

If Case 2. Occurs, it follows from Lemma 2.3.(a) that

xp+1 > 1, xp+2 < 1, xp+3 < 1, xp+4 > 1, xp+5 < 1, xp+6 < 1, xp+7 > 1, xp+8 < 1, xp+9 < 1, . . .

It means that the rule of the lengths of positive and negative semi-cycles of the solutionof Eq.(1.1) to occur successively is . . . 2−, 1+, 2−, 1+, 2−, 1+ . . .

Therefore, the proof is complete. �

Now we present the global asymptotically stable results for Eq.(1.1).

3.2. Theorem. Assume that a ∈ (0,∞). Then the positive equilibrium of Eq.(1.1) isglobally asymptotically stable.

Dynamics of a Nonlinear Rational Difference Equation 13

Proof. We should prove that the positive equilibrium point x of Eq.(1.1) is both locallyasymptotically stable and globally attractive. The linearized equation of Eq.(1.1) aboutthe positive equilibrium point x = 1 is

yn+1 = 0.yn +2

3.yn−2 + 0.yn−3 , n = 0, 1, ...

By virtue of [7, Remark 1.3.7], x is locally asymptotically stable. It remains to verifythat every positive solution {xn}∞n=−3 of Eq.(1.1) converges to 1 as n→∞. Namely, wewant to prove

(3.1) limn→∞

xn = 1

If the solution is nonoscillatory about the positive equilibrium point x of Eq.(1.1), thenfrom Lemma 2.1 and Lemma 2.4, the solution is either equal to 1 or eventually positiveor negative one which has an infinite number of terms and monotonically tends to thepositive equilibrium point x of Eq.(1.1), and so Eq.(3.1) holds.Therefore, it suffices toprove that Eq.(3.1) holds for the solution to be strictly oscillatory.

Consider now {xn} to be strictly oscillatory about the positive equilibrium point x ofEq.(1.1). By virtue of Theorem 3.1, one understands that the rule for the lengths ofpositive and negative semi-cycles which occur successively is . . . 2+, 1−, 2+, 1−, 2+, 1−, . . .or . . . , 2−, 1+, 2−, 1+, 2−, 1+, . . .

Now, we investigate the case where the rule for the lengths of positive and negativesemi-cycles which occur successively is . . . 2+, 1−, 2+, 1−, . . .

For simplicity, we denote by {xt, xt+1}+ the terms of a positive semi-cycle of lengthtwo, followed by {xt+2}− the terms of a negative semi-cycle with length one,followed by{xt+3, xt+4}+ the terms of a positive semi-cycle of length two, followed by {xt+5}− theterms of a negative semi-cycle with length one,and so on. Namely, the rule for the lengthsof positive and negative semi-cycles to occur successively can be periodically expressedas follows for n = 0, 1, . . .:

{xt+6n, xt+6n+1}+, {xt+6n+2}−, {xt+6n+3, xt+6n+4}+, {xt+6n+5}−

then the following results can be easily observed:

(3.2) 1 < xt+6n+4 < xt+6n+1

(3.3) 1 < xt+6n+6 < xt+6n+3 < xt+6n

(3.4) xt+6n+2 < xt+6n+5 < 1

It follows from 3.2 that {xt+6n+1}∞n=0 is decreasing with lower bound 1. So the limit

limn→∞

xt+6n+1 = L

exists and finite. Accordingly, by view of 3.2, we obtain

limn→∞

xt+6n+4 = L

Also, it is easy to see from 3.3 that {xt+6n}∞n=0 is decreasing with lower bound 1. So thelimit

limn→∞

xt+6n = M

14 S. E. Das.

exists and finite. By view of 3.4, we obtain

limn→∞

xt+6n+3 = limn→∞

xt+6n+6 = M

Lastly, from 3.4 that {xt+6n+2}∞n=0 is increasing with upper bound 1. So the limit

limn→∞

xt+6n+2 = N

exists and finite. By view of 3.4, we obtain

limn→∞

xt+6n+5 = N

Taking the limits on both sides of

xt+6n+6 =xat+6n+5xt+6n+3xt+6n+2 + xt+6n+5xt+6n+3x

at+6n+2 + 1

xat+6n+5xt+6n+2 + xt+6n+5xa

t+6n+2 + 1

one has, M = (2MNa+1 + 1)/(2Na+1 + 1), which gives rise to M = 1.

Similarly, taking the limits on both sides of

xt+6n+5 =xat+6n+4xt+6n+2xt+6n+1 + xt+6n+4xt+6n+2x

at+6n+1 + 1

xat+6n+4xt+6n+1 + xt+6n+4xa

t+6n+1 + 1

one has, N = (2NLa+1 + 1)/(2La+1 + 1), which gives rise to N = 1.Lastly, taking the limits on both sides of

xt+6n+4 =xat+6n+3xt+6n+1xt+6n + xt+6n+3xt+6n+1x

at+6n + 1

xat+6n+3xt+6n + xt+6n+3xa

t+6n + 1

one has, L = (2LMa+1 + 1)/(2Ma+1 + 1), which gives rise to L = 1.

So we can see that

limn→∞

xt+6n+k = 1, k = 0, 1, . . . , 6

For . . . , 2−, 1+, 2−, 1+, 2−, 1+, . . . can be similarly shown. �

References

[1] Agarwal R.P. Difference Equations and Inequalities, Marcel Dekker, Newyork, NY, USA, 1st

Edition, 1992.

[2] Agarwal R.P. Difference Equations and Inequalities, Marcel Dekker, Newyork, NY, USA,2nd Edition, 2000.

[3] Bayram, M. and Das, S. E.Global Asymptotic Stability of a Nonlinear Recursive Sequence,

Int. Math. For., 5(22), 1083-1089, 2010.[4] Berenhaut, K.S., Foley, J.D. and Stevic, S. The Global Attractivity of the Rational Difference

Equation, Appl. Math. Lett., 20, 54-58, 2007.[5] Chen, Y. and Li, X. Dynamical Properties in a Fourth-order Nonlinear Difference Equation,

Adv. Diff. Equ., ID.679409, 9 pages, 2010.[6] Das, S. E. Qualitative Behavior of a Fourth-order Rational Difference Equation, in review,

2010.

[7] Das, S. E. Global Asymptotic Stability for a Fourth-order Rational Difference Equation, Int.

Math. For., 5(32), 1591-1596, 2010.[8] Kocic, V. L. and Ladas, G. Global Behavior of Nonlinear Difference Equations of Higher

Order with Applications, Vol. 256 of Mathematics and its Applications, Kluwer AcademicPublishers, Dordrecht, The Netherlands, 1993.

[9] Li, X. and Zhu, D. Global asymptotic stability of a nonlinear recursive sequence,

Compt.Math.Appl., 17, 833- 838, 2004.

[10] Li, X. Qualitative properties for a fourth-order rational difference equation,J.Math.Anal.Appl., 311, 103 - 111, 2005.


ON THE SPECTRAL NORMS OF TOEPLITZ

MATRICES WITH FIBONACCI AND

LUCAS NUMBERS

Suleyman Solak∗† and Mustafa Bahsi‡


Abstract

This paper is concerned with the work of the authors’ [M.Akbulakand D. Bozkurt, on the norms of Toeplitz matrices involving Fibonacciand Lucas numbers, Hacettepe Journal of Mathematics and Statistics,37(2), (2008), 89-95] on the spectral norms of the matrices: A = [Fi−j ]and B = [Li−j ], where F and L denote the Fibonacci and Lucas num-bers, respectively. Akbulak and Bozkurt have found the inequalities forthe spectral norms of n×n matrices A and B, as for us, we are findingthe equalities for the spectral norms of matrices A and B.

Keywords: Spectral norm, Toeplitz matrix, Fibonacci number, Lucas Number.

2000 AMS Classification: 15A60, 15A15, 15B05, 11B39.

1. Introduction and Preliminaries

The matrix T = [tij ]n−1i,j=0 is called Toeplitz matrix such that tij = tj−i. In Section 2,

we calculate the spectral norms of Toeplitz matrices

(1) A = [Fj−i]n−1i,j=0

and

(2) B = [Lj−i]n−1i,j=0

where Fk and Lk denote k-th the Fibonacci and Lucas numbers, respectively.Now we start with some preliminaries. Let A be any n × n matrix. The spectral

norm of the matix A is defined as ‖A‖2 =√

max1≤i≤n

|λi (AHA)| where AH is the conju-

gate transpose of matrix A. For a square matrix A, the square roots of the eigenvalues

∗N.E. University, A.K. Education Faculty, 42090, Meram, Konya-TURKEY. E-mail:[email protected]†Corresponding Author.‡Aksaray University, Education Faculty, Aksaray-TURKEY. E-mail: [email protected]

16 S. Solak, M. Bahsi

of AHA are called singular values of A. Generally, we denote the singular values asσn =

{√λi : λi is eigenvalue of matrix AHA

}. Moreover, the spectral norm of matrix A

is the maximum singular value of matrix A. The equation det(A− λI) = 0 is known asthe characteristic equation of matrix A and the left-hand side known as the character-istic polynomial of matrix A. The solutions of characteristic equation are known as theeigenvalues of matrix.

Fibonacci and Lucas numbers are the numbers in the following sequences, respectively:

0, 1, 1, 2, 3, 5, 8, 13, 21, . . . and 2, 1, 3, 4, 7, 11, 18, 29, 47, . . .

in addition, these numbers are defined backwards by

0, 1,−1, 2,−3, 5,−8, 13,−21, . . . and 2,−1, 3,−4, 7,−11,−18, 29,−47, . . .

2. Main Results

2.1. Theorem. Let the matrix A be as in (1). Then the singular values of A are

σ1,2 =

{Fn, if n is even√F 2n − 1, if n is odd

and σm = 0, where m=3,4,. . . ,n.

Proof. From matrix multiplication

AAH =

[n−1∑k=0

Fk−iFk−j

]n−1

i,j=0

.

By using mathematical induction principle on n, we have

n−1∑k=0

Fk−iFk−j =

Fn−1Fn−(i+j) + F−iF−j , if n is odd

FnFn−(i+j+1), if n is even.

Since the singular values of matrix A are the square roots of the eigenvalues of matrixAAH , we must find the roots of characteristic equation

∣∣λI −AAH∣∣ = 0, for this there

are two cases.

Case I: If n is odd, since AAH =[Fn−1Fn−(i+j) + F−iF−j

]n−1

i,j=0, in this case the

characteristic equation:

∣∣∣λI −AAH∣∣∣ =

∣∣∣∣∣∣∣∣∣λ− Fn−1Fn −F 2

n−1 · · · −Fn−1F1

−F 2n−1 λ− Fn−1Fn−2 − F−1F−1 · · · −Fn−1F0 − F−1F1−n

......

...−Fn−1F1 −Fn−1F0 − F1−nF−1 · · · λ− Fn−1F−n+2 − F1−nF1−n

∣∣∣∣∣∣∣∣∣ = 0.

Let e [(i, j) , r, k] be an elementary row operation, where e [(i, j) , r, k] is addition of ktimes of addition of ith and jth rows to rth row. Firstly, we apply e [(i+ 1, i+ 2) , i,−1],(i = 1, 2, . . . , n− 2) . Secondly, we add proper times of first n− 2 rows to (n− 1)th rowand then to nth row, so we have

On the Spectral Norms of Toeplitz Matrices with Fibonacci. . . 17

∣∣∣λI −AAH∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∣

λ −λ −λ 0 · · · 0 0 00 λ −λ −λ · · · 0 0 0...

......

......

......

0 0 0 0 · · · λ −λ −λ0 0 0 0 · · · 0 λ− F 2

n + 1 00 0 0 0 · · · 0 0 λ− F 2

n + 1

∣∣∣∣∣∣∣∣∣∣∣∣∣= 0

= λn−2 (λ− F 2n + 1

)2= 0.

Hence, the singular values of the matrix A are

σ1,2 = F 2n − 1, σm = 0, where m = 3, 4, . . . , n.

Case II : If n is even, since AAH =[FnFn−(i+j+1)

]n−1

i,j=0, the characteristic equation:

∣∣∣λI −AAH∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣

λ− FnFn−1 −FnFn−2 · · · −FnF1 −FnF0

−FnFn−2 λ− FnFn−3 · · · −FnF0 −FnF−1

......

......

−FnF1 −FnF0 · · · λ− FnF3−n −FnF2−n

−FnF0 −FnF−1 · · · −FnF2−n λ− FnF1−n

∣∣∣∣∣∣∣∣∣∣∣= 0.

If we apply elemanter row operations in Case I to the determinant given above, we have

∣∣∣λI −AAH∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∣

λ −λ −λ 0 · · · 0 0 00 λ −λ −λ · · · 0 0 0...

......

......

......

0 0 0 0 · · · λ −λ −λ0 0 0 0 · · · 0 λ− F 2

n 00 0 0 0 · · · 0 0 λ− F 2

n

∣∣∣∣∣∣∣∣∣∣∣∣∣= 0

= λn−2 (λ− F 2n

)2= 0.

In that case, the singular values of the matrix A are

σ1,2 = F 2n , σm = 0, where m = 3, 4, . . . , n.

Thus the proof is completed. �

2.2. Corollary. Let the matrix A be as in (1), then ‖A‖2 =

{Fn, if n is even√F 2n − 1, if n is odd

.

Proof. The proof is trivial from Theorem 2.1. �

2.3. Theorem. Let the matrix B be as in (2). Then the singular values of B are

σ1,2 =

{Ln ± 1, if n is odd√F 2n − 1, if n is even

and σm = 0, where m = 3, 4, . . . , n.

Proof. From matrix multiplication

BBH =

[n−1∑k=0

Lk−iLk−j

]n−1

i,j=0

.

18 S. Solak, M. Bahsi

By using mathematical induction principle on n, we have

n−1∑k=0

Lk−iLk−j =

Fn−(i+j+1)Ln−1 + Fn−(i+j+2)Ln+2 − 5F−iF−j , if n is odd

5FnFn−(i+j+1), if n is even.

Firstly, we must find the roots of characteristic equation∣∣λI −BBH

∣∣ = 0, for this thereare two cases.

Case I: If n is odd, sinceBBH =[Fn−(i+j+1)Ln−1 + Fn−(i+j+2)Ln+2 − 5F−iF−j

]n−1

i,j=0,

in this case the characteristic equation:

∣∣∣λI −BBH∣∣∣ =

∣∣∣∣∣∣∣∣∣λ− Fn−1Ln−1 − Fn−2Ln+2 · · · −F0Ln−1 − F−1Ln+2

−Fn−2Ln−1 − Fn−3Ln+2 · · · −F−1Ln−1 − F−2Ln+2 + 5F−1F1−n

......

−F0Ln−1 − F−1Ln+2 · · · λ− F1−nLn−1 − F−nLn+2 + 5F1−nF1−n

∣∣∣∣∣∣∣∣∣ = 0.

If we apply elementary row operations in Case I of Theorem 2.1 to the determinant givenabove, we have

∣∣∣λI −BBH∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∣

λ −λ −λ 0 · · · 0 0 00 λ −λ −λ · · · 0 0 0...

......

......

......

0 0 0 0 · · · λ −λ −λ0 0 0 0 · · · 0 λ− a1 2Fn−3Ln

0 0 0 0 · · · 0 −2Fn−1Ln λ− a2

∣∣∣∣∣∣∣∣∣∣∣∣∣= 0

= λn−2 [λ2 −((Ln − 1)2 + (Ln + 1)2

)λ+ (L2

n − 1)2]

= 0

where a1 = (Ln − 1)2 − (2Fn−2 − 2)Ln and a2 = (Ln + 1)2 + (2Fn−2 − 2)Ln. Hence, thesingular values of the matrix B are

σ1,2 = Ln ± 1, σm = 0, where m = 3, 4, . . . , n.

Case II: If n is even, since BBH =[5FnFn−(i+j+1)

]n−1

i,j=0, in this case the characteristic

equation:

∣∣∣λI −BBH∣∣∣ =

∣∣∣∣∣∣∣∣∣λ− 5FnFn−1 −5FnFn−2 · · · −5FnF0

−5FnFn−2 λ− 5FnFn−3 · · · −5FnF−1

......

...−5FnF0 −5FnF−1 · · · λ− 5FnF1−n

∣∣∣∣∣∣∣∣∣ = 0.

If we apply elementary row operations in Case I of Theorem 2.1 to the determinant givenabove, we have

On the Spectral Norms of Toeplitz Matrices with Fibonacci. . . 19

∣∣∣λI −BBH∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∣

λ −λ −λ 0 · · · 0 0 00 λ −λ −λ · · · 0 0 0...

......

......

......

0 0 0 0 · · · λ −λ −λ0 0 0 0 · · · 0 λ− L2

n + 4 00 0 0 0 · · · 0 0 λ− L2

n + 4

∣∣∣∣∣∣∣∣∣∣∣∣∣= 0

= λn−2 (λ− L2n + 4

)2= 0.

Hence, the singular values of the matrix B are

σ1,2 =√L2

n − 4, σm = 0, where m = 3, 4, . . . , n.

Thus the proof is completed. �

2.4. Corollary. Let the matrix B be as in (2), then ‖B‖2 =

{Ln + 1, if n is odd√L2

n − 4, if n is even.

Proof. The proof is trivial from Theorem 2.3. �

References

[1] Akbulak, M., and Bozkurt, D. On the norms of Toeplitz matrices involving Fibonacci and

Lucas numbers, Hacettepe Journal of Mathematics and Statistic, 37 (2), 89-95, 2008.

Hacettepe Journal of Mathematics and StatisticsVolume 42(1) (2013), 21 – 29

FIXED POINTS OF MULTIVALUED

MAPPING SATISFYING CIRIC TYPE

CONTRACTIVE CONDITIONS IN

G-METRIC SPACES

Mujahid Abbas ∗ † , Talat Nazır ∗ § and B. E. Rhoades‡


Abstract

In this paper, study of necessary conditions for existence of fixed pointof multivalued mappings satisfying Ciric type contractive conditionsin the setting of generalized metric spaces is initiated. Examples tosupport our results are presented. Since every symmetric generalizedmetric reduces to an ordinary metric, we give a new example of a non-symmetric generalized metric to justify the study of fixed point theoryin generalized metric spaces.

Keywords: Multivalued mappings, fixed point, non symmetric, generalized metricspaces.

2000 AMS Classification: 47H10, 54H25, 54E50.

1. Introduction and Preliminaries

The study of fixed points of mappings satisfying certain contractive conditions hasbeen at the center of rigorous research activity. Mustafa and Sims [10] generalized theconcept of a metric space. Based on the notion of generalized metric spaces, Mustafaet al. ([9, 11, 12]) obtained some fixed point theorems for mappings satisfying differentcontractive conditions. Abbas and Rhoades [1] motivated the study of common fixedpoint theory in generalized metric spaces. Recently, Saadati et al. [14] proved some fixed

∗Department of Mathematics, COMSATS Institute of Information Technology, 22060, Ab-

bottabad, Pakistan.†E-mail: (M. Abbas) [email protected]§E-mail: (T. Nazır) [email protected]‡Department of Mathematics, Indiana University, Bloomington, IN 47405-7106.

E-mail: (B. E. Rhoades) [email protected]

22 M. Abbas, T. Nazır, B. E. Rhoades

point results for contractive mappings in partially ordered G− metric spaces. Abbas etal. [2] obtain some periodic point results in generalized metric spaces.

The aim of this paper is to prove various fixed points results for multivalued mappingstaking closed values in generalized metric spaces. It is worth mentioning that our resultsdo not rely on the notion of continuity of the mappings involved therein. Our resultsextend and unify various comparable results in ([4], [5] and [13]).Consistent with Mustafa and Sims [10], the following definitions and results will be neededin the sequel.

1.1. Definition. Let X be a nonempty set. Suppose that a mapping G : X ×X ×X →R+ satisfies:

(a) G(x, y, z) = 0 if x = y = z;(b) 0 < G(x, y, z) for all x, y ∈ X, with x 6= y;(c) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X, with y 6= z;(d) G(x, y, z) = G(p{x, y, z}), where p is a permutation of x, y, z (symmetry);(e) G(x, y, z) ≤ G(x, a, a) +G(a, y, z) for all x, y, z, a ∈ X.

Then G is called a G− metric on X and (X,G) is called a G− metric space.

1.2. Definition. A sequence {xn} in a G− metric space X is:

(i) a G− Cauchy sequence if, for any ε > 0, there is an n0 ∈ N ( the set naturalnumber ) such that for all n,m, l ≥ n0, G(xn, xm, xl) < ε,

(ii) a G− Convergent sequence if, for any ε > 0, there is an x ∈ X and an n0 ∈ N,such that for all n,m ≥ n0, G(x, xn, xm) < ε.

A G− metric space on X is said to be G− complete if every G− Cauchy sequence inX is G− convergent in X. It is known that {xn} G− converges to x ∈ X if and only ifG(xm, xn, x)→ 0 as n,m→∞ [10].

1.3. Proposition. [10] Let X be a G− metric space. Then the following are equivalent:

(1) {xn} is G− convergent to x.(2) G(xn, xn, x)→ 0 as n→∞.(3) G(xn, x, x)→ 0 as n→∞.(4) G(xn, xm, x)→ 0 as n,m→∞.

1.4. Definition. A G− metric on X is said to be symmetric if G(x, y, y) = G(y, x, x)for all x, y ∈ X.

1.5. Proposition. Every G− metric on X will define a metric dG on X by

dG(x, y) = G(x, y, y) +G(y, x, x), ∀ x, y ∈ X.

For a symmetric G− metric space

dG(x, y) = 2G(x, y, y), ∀ x, y ∈ X.

However, if G is not symmetric, then the following inequality holds:

3

2G(x, y, y) ≤ dG(x, y) ≤ 3G(x, y, y), ∀ x, y ∈ X.

Now we give an example of non-symmetric G− metric.

1.6. Example. Let X = {1, 2, 3}, G : X ×X ×X → R+, be defined as

Fixed Points of Multivalued Mapping Satisfying Ciric Type . . . 23

(x, y, z) G(x, y, z)

(1, 1, 1), (2, 2, 2), (3, 3, 3) 0

(1, 1, 2), (1, 2, 1), (2, 1, 1),(2, 2, 3), (2, 3, 2), (3, 2, 2),(1, 1, 3), (1, 3, 1), (3, 1, 1),(1, 2, 2), (2, 1, 2), (2, 2, 1),(2, 3, 3), (3, 2, 3), (3, 3, 2)

1

(1, 2, 3), (1, 3, 2), (2, 1, 3),(2, 3, 1), (3, 1, 2), (3, 2, 1),(1, 3, 3), (3, 1, 3), (3, 3, 1)

2

Note that G satisfies all of the axioms of a generalized metric but G(1, 1, 3) 6= G(1, 3, 3).Therefore G is not a symmetric on X.Let X be a G− metric space. We denote by P (X) the family of all nonempty subsets ofX, and by Pcl(X) the family of all nonempty closed subsets of X.A point x in X is called a fixed point of a multivalued mapping T : X → Pcl(X) providedx ∈ Tx. The collection of all fixed point of T is denoted by Fix(T ).

2. Fixed Point Theorems

Kannan [4] proved a fixed point theorem for a single valued self mapping T of a metricspace X satisfying the property

d(Tx, Ty) ≤ h{d(x, Tx) + d(y, Ty)}for all x, y in X and for a fixed h ∈ [0, 1

2). Ciric [3] proved a fixed point theorem for a

single valued self mapping T of a metric space X satisfying the property

d(Tx, Ty) ≤ ad(x, y) + bd(x, Tx) + cd(y, Ty) + e[d(x, Ty) + d(y, Tx)]

for all x, y in X and for a fixed a, b, c, e ≥ 0 with a + b + c + 2e < 1. Latif andBeg [5] introduced the notion of a K− multivalued mapping, which is the extensionof Kannan mappings, to multivalued mappings. Continuing in this direction, Rus etal. [13] coined the term R− multivalued mapping, which is a generalization of a K−multivalued mapping.In this section, we obtain some fixed point theorems for a multivalued mapping satis-fying Ciric type contractive conditions on generalized metric spaces without using thecontinuity condition.

2.1. Theorem. Let X be a complete G− metric space and T : X → Pcl(X). If for eachx, y ∈ X, ux ∈ T (x) there exist uy ∈ T (y) such that

(2.1) G(ux, uy, uy) ≤ hmax{G(x, y, y), G(x, ux, ux), G(y, uy, uy),

1

2[G(x, uy, uy) + G(y, ux, ux)]},

where h ∈ [0, 1), then T has a fixed point.

Proof. Let x0 be an arbitrary point ofX, and x1 ∈ T (x0). Then there exists an x2 ∈ T (x1)such that

G(x1, x2, x2) ≤ hmax{G(x0, x1, x1), G(x0, x1, x1), G(x1, x2, x2),

1

2[G(x0, x2, x2) +G(x1, x1, x1)]}

= hmax{G(x0, x1, x1), G(x1, x2, x2),1

2[G(x0, x2, x2)]}.


But, from property (e) of Definition 1.1,

G(x0, x2, x2)

2≤ 1

2[G(x0, x1, x1) +G(x1, x2, x2)]

≤ max{G(x0, x1, x1), G(x1, x2, x2)},

and we now have

(2.2) G(x1, x2, x2) ≤ hmax{G(x0, x1, x1), G(x1, x2, x2)}.

If G(x1, x2, x2) = 0, then, by (1) of proposition 1 in [10], x1 = x2. Then x2 ∈ T (x1) =T (x2), and x2 is a fixed point of T.If G(x1, x2, x2) 6= 0, then (2.2) becomes

G(x1, x2, x2) ≤ hG(x0, x1, x1).

Continuing this process, we obtain a sequence {xn} in X, that is for xn ∈ T (xn−1), thereexists xn+1 ∈ T (xn) such that

G(xn, xn+1, xn+1) ≤ hmax{G(xn−1, xn, xn), G(xn−1, xn, xn), G(xn, xn+1, xn+1),

(G(xn−1, xn+1, xn+1) +G(xn, xn, xn))/2}= hmax{G(xn−1, xn, xn), G(xn, xn+1, xn+1),

(G(xn−1, xn+1, xn+1))/2}= hmax{G(xn−1, xn, xn), G(xn, xn+1, xn+1)}.

Without loss of generality we may assume that xn 6= xn+1 for each n, since, otherwise,it follows that xn+1 is a fixed point of T.

Thus we have

G(xn, xn+1, xn+1) ≤ hG(xn−1, xn, xn) ≤ ... ≤ hnG(x0, x1, x2).

For any m > n ≥ 1, repeated use of property (e) gives

G(xn, xm, xm) ≤ G(xn, xn+1, xn+1) + ...+G(xm−1, xm, xm)

≤ [hn + hn+1 + ...+ hm−1]G(x0, x1, x1) ≤ hn

1− hG(x0, x1, x1),

and so G(xn, xm, xm)→ 0 as n,m→∞. Hence {xn} is a G− Cauchy sequence. By theG− completeness of X, there exist a u ∈ X such that {xn} converges to u. Let n ≥ Nbe given. Then, for each xn ∈ T (xn−1), there exists a un ∈ T (u) such that

G(un, un, u) ≤ G(xn, un, un) +G(xn, xn, u)

≤ hmax{G(xn−1, u, u), G(xn−1, xn, xn), G(u, un, un),

1

2[G(xn−1, un, un) +G(u, xn, xn)]}+G(xn, xn, u)

≤ hmax{G(xn−1, u, u), G(xn−1, xn, xn), G(un, un, u),

1

2[G(xn−1, u, u) +G(un, un, u) +G(xn, xn, u)]}+G(xn, xn, u).

Now, if

max{G(xn−1, u, u), G(xn−1, xn, xn), G(un, un, u),

1

2[G(xn−1, u, u) +G(un, un, u) +G(u, xn, xn)]}

= G(xn−1, u, u),

implies that

G(un, un, u) ≤ hG(xn−1, u, u) +G(xn, xn, u).


Taking limit as n→∞, implies G(un, un, u)→ 0, and un → u.If


1

2[G(xn−1, u, u) +G(un, un, u) +G(xn, xn, u)]}

= G(xn−1, xn, xn),

then

G(un, un, u) ≤ hG(xn−1, xn, xn) +G(xn, xn, u)

≤ hG(xn−1, u, u) + 2G(xn, xn, u).

On letting limit n→∞, implies G(un, un, u)→ 0, and un → u.In case


1


= G(un, un, u),

then

G(un, un, u) ≤ hG(un, un, u) +G(xn, xn, u)

which further implies that

G(un, un, u) ≤ 1

1− hG(xn, xn, u).

Taking the limit as n→∞, implies G(un, un, u)→ 0, gives un → u.Finally, if


1


=1

2[G(xn−1, u, u) +G(un, un, u) +G(xn, xn, u)],

then

G(un, un, u)

≤ h

2[G(xn−1, u, u) +G(un, un, u) +G(xn, xn, u)] +G(xn, xn, u)

≤ h

2G(xn−1, u, u) +

1

2G(u, un, un) +

3

2G(xn, xn, u),

which further implies

G(un, un, u) ≤ hG(xn−1, u, u) + 3G(xn, xn, u).

Taking the limit as n→∞, implies that G(un, un, u)→ 0.Thus un → u as n→∞. Since un ∈ T (u) and T (u) is closed, it follows that u ∈ T (u). �

The following corollary generalizes Theorem 3.1 of Rus et al. [13] to G− metric spaces.

2.2. Corollary. Let X be a complete G− metric space and T : X → Pcl(X). If for eachx, y ∈ X, ux ∈ T (x), there exists a uy ∈ T (y) such that

(2.3) G(ux, uy, uy) ≤ a1G(x, y, y) + a2G(x, x, y) + a3G(x, ux, ux)

+ a4G(x, x, ux) + a5G(y, uy, uy) + a6G(y, y, uy),


where ai ≥ 0 for i = 1, 2, ..., 6 and a1 + a3 + a5 + 2(a2 + a4 + a6) < 1, then T has a fixedpoint.

Proof. Note that (2.3) implies that

G(ux, uy, uy) ≤ hmax{G(x, y, y), G(x, ux, ux), G(y, uy, uy),

G(x, x, y)

2,G(x, x, ux)

2,G(y, y, uy)

2},

where h = a1 + a3 + a5 + 2(a2 + a4 + a6) < 1.Which further implies that

G(ux, uy, uy) ≤ hmax{G(x, y, y), G(x, ux, ux), G(y, uy, uy)},

and the result follows from Theorem 2.1. �

2.3. Example. Let X = [0,∞) and G(x, y, z) = max{|x− y| , |y − z| , |z − x|} be asymmetric G−metric on X. Define T : X → Pcl(X) as

Tx = [0,x

6].

Now for case x = y, ux ∈ Tx. Take uy = 0, then

G(ux, uy, uy)

= ux ≤x

6

≤ 2

12(0) +

3

12(5x

6) +

3

12(x)

≤ 1

12(x− y) +

1

12(x− y) +

2

12(x− ux) +

1

12(x− ux) +

2

12(y − uy) +

1

12(y − uy)

= a1G(x, y, y) + a2G(x, x, y) + a3G(x, ux, ux)

+a4G(x, x, ux) + a5G(y, uy, uy) + a6G(y, y, uy).

Thus (2.3) is satisfied with a1 + a3 + a5 + 2(a2 + a4 + a6) =11

12.

Now when x < y, ux ∈ Tx. Take uy = 0, then

G(ux, uy, uy)

= ux ≤x

6

≤ 2

12(0) +

3

12(5x

6) +

3

12(x)

≤ 1

12(y − x) +

1

12(y − x) +

2

12(x− ux) +

1

12(x− ux) +

2

12(y − uy) +

1

12(y − uy)




12.


Finally for, y < x, ux ∈ Tx. Take uy =y

6, then

G(ux, uy, uy)

= |ux − uy| ≤ ux + uy ≤1

6(x+ y)

≤ 2

12(x− y) +

3

12(5x

6) +

3

12(5y

6)

≤ 1

12(x− y) +

1

12(x− y) +

2

12(x− ux) +

1

12(x− ux) +

2

12(y − uy) +

1

12(y − uy)




12.

Hence all conditions of Corollary 2.2 are satisfied. Moreover, T has a fixed point.

2.4. Corollary. Let X be a complete G− metric space and T : X → Pcl(X). If for eachx, y ∈ X, ux ∈ T (x), there exist uy ∈ T (y) such that

(2.4) G(ux, uy, uy) ≤ αG(x, y, y) + βG(x, ux, ux) + γG(y, uy, uy),

where α, β, γ ≥ 0 and α+ β + γ < 1, then T has a fixed point.

2.5. Example. Let X = {0, 1} and a nonsymmetric G− metric from X to R+ be defineas

G(0, 0, 0) = G(1, 1, 1) = 0,

G(0, 0, 1) = G(0, 1, 0) = G(1, 0, 0) = 0.5,

G(0, 1, 1) = G(1, 0, 1) = G(1, 1, 0) = 1.

Define T : X → Pcl(X) as

T (0) = T (1) = {0, 1}.

Now if x = 0, y = 0, ux ∈ T (0). Then two cases arise.When ux = 0, take uy = 0 ∈ T (y), then

G(ux, uy, uy) = G(0, 0, 0) = 0

=1

8(0) +

3

8(0) +

3

8(0)

= αG(0, 0, 0) + βG(0, 0, 0) + γG(0, 0, 0)

= αG(x, y, y) + βG(x, ux, ux) + γG(y, uy, uy).

When ux = 1, take uy = 1 ∈ T (y), then

G(ux, uy, uy) = G(1, 1, 1) = 0

<1

8(0) +

3

8(1) +

3

8(1)

= αG(0, 0, 0) + βG(0, 1, 1) + γG(0, 1, 1)


Thus (2.4) is satisfied with α+ β + γ =7

8.


For case x = 0, y = 1, ux ∈ T (0). Then for ux = 0, take uy = 0 ∈ T (y), then

G(ux, uy, uy) = 0

<1

8(1) +

3

8(0) +

3

8(0.5)

= αG(0, 1, 1) + βG(0, 0, 0) + γG(1, 0, 0)


And when ux = 1, take uy = 0 ∈ T (y), then

G(ux, uy, uy) = G(1, 0, 0) = 0.5

<1

8(1) +

3

8(1) +

3

8(0.5)

= αG(0, 1, 1) + βG(0, 1, 1) + γG(1, 0, 0)



8.

For case x = 1, y = 0, ux ∈ T (1). Then for ux = 0, take uy = 0, we have

G(ux, uy, uy) = 0

<1

8(0.5) +

3

8(0.5) +

3

8(0)

= αG(1, 0, 0) + βG(1, 0, 0) + γG(0, 0, 0)


and when ux = 1, again by taking uy = 1, we have

G(ux, uy, uy) = G(1, 1, 1) = 0

<1

8(0.5) +

3

8(0) +

3

8(1)

= αG(1, 0, 0) + βG(1, 1, 1) + γG(0, 1, 1)



8.

Finally for x = 1, y = 1, ux ∈ T (1), then for the case ux = 0, take uy = 0 ∈ T (1), wehave

G(ux, uy, uy) = 0

<1

8(0) +

3

8(0.5) +

3

8(0.5)

= αG(1, 1, 1) + βG(1, 0, 0) + γG(1, 0, 0)


And if ux = 1, take uy = 1 ∈ T (1), implies

G(ux, uy, uy) = G(1, 1, 1) = 0

=1

8(0) +

3

8(0) +

3

8(0)

= αG(1, 1, 1) + βG(1, 1, 1) + γG(1, 1, 1)



8. Hence all the conditions of Corollary 2.4 are

satisfied and Fix(T ) 6= 0.

The following corollary generalizes Theorem 4.1 of Latif and Beg [5] to G− metric Spaces.



G(ux, uy, uy) ≤ h[G(x, ux, ux) +G(y, uy, uy)],

where 0 ≤ h < 1, then T has a fixed point.


G(ux, uy, uy) ≤ λG(x, y, y),

where 0 ≤ λ < 1, then T has a fixed point.

References

[1] Abbas, M. and Rhoades, B. E. Common fixed point results for non-commuting mappingswithout continuity in generalized metric spaces, Appl. Math. and Computation 215 , 262–

269, 2009.

[2] Abbas, M., Rhoades, B. E. and Nazir T. Some periodic point results in generalized metricspaces, Appl. Math. and Computation 217 , 4094-4099, 2010.

[3] Ciric, Lj.Generalized contractions and fixed-point theorems, Publ. Inst. Math. 12 (26), 19-26,

1971.[4] Kannan, R. Some results on fixed points, Bull. Calcutta, Math. Soc. 60, 71–76, 1968.

[5] Latif, A. and Beg, I. Geometric fixed points for single and multivalued mappings, Demons.

Math., 30 (4), 791–800, 1997.[6] Matkowski, J. Fixed point theorems for mappings with a contractive iterate at a point, Pro-

ceedings of the American Mathematical Society, 62(2), 344-348, 1977.

[7] Rus, I. A., Petrusel A. and Sintamarian A. Data dependence of fixed point set of somemultivalued weakly Picard operators, Nonlinear Analysis 52, 1944-1959, 2003.

[8] Saadati, R., Vaezpour, S. M., Vetro, P. and Rhoades, B. E. Fixed point theorems in generalizedpartially ordered G− metric spaces, Mathematical and Computer Modelling, in press, 2010.

[9] Mustafa, Z. and Sims, B. Some remarks concerning D− metric spaces, Proc. Int. Conf. on

Fixed Point Theory and Applications, Valencia (Spain), 189–198, 2003.[10] Mustafa, Z. and Sims, B. A new approach to generalized metric spaces, J. of Nonlinear and

Convex Analysis, 7 (2), 289–297, 2006.

[11] Mustafa, Z., Obiedat, H. and Awawdehand, F. Some fixed point theorem for mapping oncomplete G− metric spaces, Fixed Point Theory Appl., 2008, Article ID 189870, 12 pages.

[12] Mustafa, Z. and Sims, B. Fixed point theorems for contractive mapping in complete G−metric spaces, Fixed Point Theory Appl., Article ID 917175, 10 pages, 2009.

[13] Rus, I. A. , Petrusel, A. and Sintamarian, A. Data dependence of fixed point set of some

multivalued weakly Picard operators, Nonlinear Analysis 52, 1944–1959, 2003.

[14] Saadati, R., Vaezpour, S. M., Vetro, P. and Rhoades, B. E. Fixed point theorems in gener-alized partially ordered G− metric spaces, Math. and Comp. Modelling, 52 (5-6), 797–801,2010.


Cofibration Category and Homotopies of

Three–Crossed Complexes


Hasan Atik∗

Abstract

In this work, we show that category of totally free 2–crossed complexesand that of totally free 3–crossed complexes are cofibration categoriesin the sense of Baues ([4]). We also explore homotopies for 3–crossedmodules and 3–crossed complex morphisms.

1. Introduction

Crossed modules were first defined by Whitehead in [15]. They model homotopyconnected 2–types. Conduche ([9]), in 1984, described the notion of 2–crossed modulesas a model for homotopy connected 4–types. Eventually, Arvasi, Uslu and Kuzpinari ([2])introduced 3–crossed modules as a model for homotopy connected 4–types. The definitionof a homotopy of morphisms of crossed complexes is well known due to Whitehead andthis was put in the general context of crossed complexes (of groupoids) by Brown andHiggins in [6]. Also homotopies for 2–crossed complexes can be found in Martin’s work[11]. By following Martin’s method we give homotopies for 3–crossed complexes.

T.Porter explains cofibration category as follows: The notion of cofibration categorywas introduced by Hans–Joachim Baues as a variant of the category of cofibrant objects,(for which, see category of fibrant objects and dualise). The axioms are substantiallyweaker than those of Quillen’s model category [13], but add one axiom to those of K. S.Brown. In the first chapter of his book, Algebraic Homotopy, Baues suggests two criteriafor an axiom system:

1. The axioms should be sufficiently strong to permit the basic constructions of ho-motopy theory. 2. The axioms should be as weak (and as simple) as possible, so that theconstructions of homotopy theory are available in as many contexts as possible.

Baues in [3] has shown that category of totally free crossed complexes and categoryof totally free quadratic complexes are cofibration category.

In this article, we obtain similar results. We show that the category of totally free2–crossed complexes is a cofibration category and we define homotopies for morphisms of2–crossed complexes. Then we get the following result: Homotopy classes of category of

∗Meliksah University, Science and Art Faculty, Mathematics Department, Kayseri, TURKEYe–Mail: [email protected]

32 H. Atik

totally free 2–crossed complexes is equivalent to the localization of 2–crossed complexeswith respect to weak equivalences.

2. Preliminaries

2.1. Definition. A Baues “cofibration category” is a category (C, cof, we) consistingof a category C and two distinguished classes of maps cof and we, called “cofibrations”and “weak equivalences” respectively. A map in C is a trivial cofibration if it is both aweak equivalence and a cofibration. Maps in C are subject to the axioms below:

BCF1: All isomorphisms of C are trivial cofibrations. Cofibrations are stable undercomposition.

BCF2: (Two out of three axiom) If f, g are maps of C such that gf is defined, if anytwo of f, g, gf are weak equivalences, then so is the third.

BCF3: (Push out axiom) Given a solid diagram

Af // A ∪B Y

B

i

OO

f// Y

i

OO

in C, with i being a cofibration, then the pushout exists in C and i is a cofibration.Moreover:

(a) if f is a weak equivalence, so is f,(b) if i is a weak equivalence, so is i.

BCF4: (Factorization axiom) Any map of C admits a factorization as a cofibrationfollowed by a weak equivalence.

BCF5: (Axiom on fibrant models) For each object X in C there is a trivial cofibrationX → RX where RX is fibrant in C. An object R in a cofibration category is fibrant ifeach trivial cofibration i : R → Q admits a retraction r : Q → R such that ri = 1. Wecall X → RX a fibrant model of X; if X is fibrant we take RX = X.

2.2. Definition. We call (C, cof, we) a“cofibration structure” if all axioms of cofibrationcategory are satisfied except the axiom BCF3(a).

Hence a cofibration structure which satisfies BCF3(a) is a cofibration category. Forexample, let (C, cof, fib, we) be a model category in the sense of Quillen, then (C, cof, we)is a cofibration structure. An object X in a category C is cofibrant if ∗ → X is acofibration where ∗ is the initial object. A full subcategory of a category C consisting ofcofibrant objects is denoted by Cc.

3. Cofibrations in the Category of 2–crossed Complexes

The following definition of 2–crossed module is equivalent to that given by Conduche.A 2–crossed module of groups consists of a complex of groups

L∂2 // M

∂1 // N

together with (a) actions of N on M and L so that ∂2, ∂1 are morphisms of N–groups,and (b) an N–equivariant function

{ , } : M ×M −→ L,

Cofibration Category and Homotopies of Three–Crossed Complexes 33

called a Peiffer lifting. This data must satisfy the following axioms:

2CM1) ∂2{m,m′} = mm′m−1(∂1mm′−1

)2CM2) {∂2l, ∂2l′} = [l′, l]

2CM3) (i) {mm′,m′′} = ∂1m{m′,m′′}{m,m′m′′m′−1}(ii) {m,m′m′′} = {m,m′}mm

′m−1

{m,m′′}2CM4) {m, ∂2l}{∂2l,m} = ∂1mll−1

2CM5) n{m,m′} = {nm,nm′}

for all l, l′ ∈ L, m,m′,m′′ ∈M and n ∈ N .

3.1. Definition. [12] A 2–crossed complex C = {Cn, dn, { , }} is a diagram

· · ·d5 // C4

d4 // C3d3 // C2

d2 // C1d1 // C0

of homomorphisms between groups such that dn−1dn = 1 for n ≥ 2 and such that thefollowing properties are satisfied. The ({ , }, d2, d1) is a 2–crossed module. MoreoverCn is a right π–module.

A map f : C → C′ between 2–crossed complexes is a family of homomorphismsbetween groups for n ≥ 1

fn : Cn → C′n with fn−1dn = dnfn

such that (f3, f2, f1) is a map between 2–crossed modules.

Let X2Comp be the category of 2–crossed complexes and maps, we define the homo-topy groups

π1(C) = π = coker(d1)

πn(C) =ker dn

imdn+1, n ≥ 1.

A map fn is a weak equivalence if πn(fn) is an isomorphism for n ≥ 1. We call a2–crossed complex C totally free if C0 is a free group, d1 : C1 → C0 is a free pre–crossedmodule, and d2 : C2 → C1 is given by a free 2–crossed module. Totally free objectsform a full subcategory of 2–crossed complexes. We denote it by FreeX2. Clearly,FreeX2 ⊂ X2Comp.

3.2. Definition. A map f : A→ B in 2–crossed complexes is a cofibration if f is a freeextension in each degree n for n ≥ 1.

Here we say that f is a free extension in degree n with basis ∂n : Xn → Bn−1

where Xn is a set and a map j : Xn → Bn with dnj = ∂n which satisfy the followinguniversal property. Let B′ be any 2–crossed complex, An, Bn, B′n be n–skeleton of

A,B,B′ and let b : A → B′, an−1 : Bn−1 → B′n−1

be 2–crossed complex maps with

an−1fn−1 = bn−1 : An−1 → B′n−1

and assume a function c : Xn → B′n is chosen such

34 H. Atik

that the following diagram of unbroken arrows commutes.

An

dn

��

fn

//

bn

))Bn

an //

dn

��

B′n

dn

��

Xn

j

bbEEEEEEEE

∂n

||yyyyyyyyy

c

<<yyyyyyyy

An−1

fn−1 //

bn−1

55Bn−1 an−1

// B′n−1

Then there is a unique 2–crossed complex map a : Bn → B′n

for which an extends thediagram commutatively. It is clear that cofibrant objects in 2–crossed complexes areexactly the totally free 2–crossed complexes. Then

FreeX2 = X2Compc

where X2Compc denotes full subcategory of category of 2–crossed complexes consistingof cofibrant objects. The next lemma shows that free extension in each degree exists.

3.3. Lemma. Let An be an n–skeleton and assume fn−1 : An−1 → Bn−1 and a function∂n : Xn → Bn−1are given. Then a free extension f : An → Bn with basis ∂n existsprovided that dn−1∂n = 1.

Proof is analogue to the case of free extensions in the quadratic complexes in [3].

3.4. Theorem. The category of 2–crossed complexes with cofibrations and weak equiva-lences is a cofibration structure for which all objects are fibrant.

Proof. We first check (BCF4). We obtain a factorization

f : Ai // B

q // C

of f : A→ C such that i is a cofibration and q is a weak equivalence.

For n = 1, B1 is free product of A1 and F (X1) where X1 is a set and F (X1) is the freegroup generated by X1. We choose X1 and q1 such that B1 → C1 → π1(C1) is surjective.

For n = 2, B′2 is free product of A′2 and F (X2) where A′2 is the B1–pre–crossed moduleinduced from the A1–pre–crossed module by the morphism of 2–crossed module A1 → B1

and F (X2) is the free pre–crossed B1–module on the set X2. Here we choose a basis X2

and ∂2 : B′2 → B1 such that ∂2(B2) is the kernel of B1 → C1 → π1(C1). Then we shouldfind q′2 such that the diagram

B′2

∂2

��

q′2 // C2

d2

��B1 q1

// C1

commutes. The map q′2 is not surjective. Therefore choose a set X ′2 such that B2 is freeproduct of B′2 and F (X ′2). Then take q2 as carrying kernel of ∂2 surjectively to kernel ofd2.

For n ≥ 3, Bn is chosen in a similar way. This completes the construction of factor-ization.


We next check that all objects are fibrant. Let a map i : A→ B be given as a trivialcofibration. We construct inductively a retraction r : B → A with ri = 1 and a homotopyα : ir ' 1relA This shows that i is actually a strong deformation retract morphism. Let

(#) i : A // Bn

rn

��gn// B

be given by the subcomplex Bn of B with (Bn)k = Bk for k ≤ n and (Bn)k = Akfor k > n. The map gn is the inclusion. We choose inductively a retraction rn and ahomotopy αn : irn ' gn relative to A. Assume rn and αn are defined by

(*) (irn)−1gn = (dαn)(αnd).

When we compose each side by d, we get

irnd = (gnd)(dαnd)−1 = d(gn(αnd)−1).

Since i is a weak equivalence we can choose a map x : Xn+1 → An+1 with dx = rnd.Moreover (ix)−1gn+1(αnd)−1 carries Xn+1 to the cycles of B by (∗). Again since i is aweak equivalence we can choose maps z : Xn+1 → An+1, y : Xn+1 → Bn+2 such that

(iz)(dy) = (ix)−1gn+1(αnd)−1.

We now define the extension rn+1 of rn by rn+1 = xz on Xn+1 and we define the extensionαn+1 of αn by αn+1 = y on Xn+1. This completes the induction.

Moreover, we construct push outs in 2–crossed complexes

Bf // B′

A

i

OO

// A′

i

OO

as follows. Let Bn be a free extension of A Then we set B′n as free extension of A′. Thebasis of B′ is given in degree n by the composition

fn−1d|Xn : Xn → An−1 → Bn−1, n ≥ 2.

The map f is the identity on Xn.Finally we prove (BCF3b). If i is a weak equivalence, then i is a strong deformation

retract by (#). This implies also that i is a strong deformation retract. In fact we define

the retraction r of i by fr in (#). And we define the homotopy α : ir ' 1relA′ by fα ongenerators. This shows that i is a weak equivalence and (BCF3b) is satisfied. �

The next lemma is given in [4].

3.5. Lemma. Let C be a cofibration structure. Then Cc with cofibrations and weakequivalences as in C is a cofibration category.

As a result of above theorem and lemmas we give the following result.

3.6. Corollary. The category of totally free 2–crossed complexes is a cofibration category

4. Homotopy of 2–crossed Modules

Recall the notion of homotopy between crossed complexes in [7]. Now similarly wedefine homotopy for 2–crossed complexes. Let

A = L∂2 // M

∂1 // N , A′ = L′∂2 // M ′

∂1 // N ′

36 H. Atik

be two 2–crossed modules and f = (f0, f1, f2) be a 2–crossed module morphism. Ahomotopy on f is a pair h = (h1, h2) of maps where h1 : N → M ′ and h2 : M → L′

satisfying equations below.

h1(nn′) = f0(n′)−1

h1(n)h1(n′)

h2(mm′) = ((f1m′)(h1∂1m

′))−1

({f1m′, f0∂1m′−1

(h1∂1m−1)}h2(m))h2(m′)

∂2h2(nm) = h1(n)f0(n)h1∂1(m−1)f0∂1(m−1)h1(n)f1(m−1)∂1h1(n)f1(n)h1∂1(m)

∂2h2(m)∂1h1(n−1)f0(n−1)

h2∂2(nl) = f0(n)f2(l−1)∂1h1(n)f2(l)∂3h2(l)∂1h1(n−1)f0(n−1)

Such a function h is called a quadratic f–derivation.

4.1. Proposition. Given a homotopy as above, the formulasf ′0(n) = f0(n)∂1h1(n)f ′1(m) = f1(m)h1∂1(m)∂2h2(m)f ′2(l) = f2(l)h2∂2(l)for all n ∈ N,m ∈M, l ∈ L define a morphism of 2–crossed modules.

We leave the proof as an exercise.

4.1. Homotopies of 2–crossed Complexes. Let A and A′ be two 2–crossed com-plexes and let f be a 2–crossed complex map A → A′. A quadratic f–derivation is asequence of maps hi : Ai → A′i+1 such that (h2, h1) is a quadratic f–derivation of 2–crossed modules and all the remaining maps are A1–equivariant for n = 3 and A1/∂A2–equivariant for n ≥ 4. We say that two 2–crossed complex maps are homotopic if thereexists a quadratic f–derivation such that

f ′1(a) = f1(a)∂2h1(a) and f ′n(a) = fn(a)(hn−1∂(a))(∂hn(a)) for n ≥ 2.

In [6] Brown and Higgins extended the notion of homotopy to n–fold homotopies. Inthis manner a 0–fold homotopy between two 2–crossed complexes B and C is simplya morphism B → C. For n ≥ 1 an n–fold homotopy B → C is a pair (h, f), wheref : B → C is a morphism of crossed complexes and h is a map of degree n from B to Ci.e., h : Bk → Ck+n. 1–fold homotopy is the homotopy we have just defined above.

Moreover, we have equivalence of categories. In [4], a corollary (IV.5.7) for quadraticcomplexes is given. Since we have homotopy relation for 2-crossed complexes and Corol-lary 3.6 then we can give analogue lemma for 2–crossed complexes.

4.2. Lemma. Homotopy classes of the category of totally free 2–crossed complexes isequivalent to the localization of 2–crossed complexes with respect to weak equivalenceswhich can be pictured as a functor M ,

M : Ho(X2Comp) −→ FreeX2/ '

For the proof, see [4]; for the localization of a category with respect to a class ofmorphisms see [13].

5. Cofibrations in the Category of 3–crossed Complexes

We follow conventions of [2] for the definition of 3–crossed modules.

5.1. Definition. A 3–crossed module consists of a complex of groups

K∂3 // L

∂2 // M∂1 // N


together with an action of N on K,L,M and an action of M on K,L and an action ofL on K so that ∂3, ∂2, ∂1 are morphisms of N,M–groups. And the M,N–equivariantliftings

{ , }(1)(0) : L× L −→ K, { , }(0)(2) : L× L −→ K,{ , }(2)(1) : L× L −→ K, { , }(1,0)(2) : M × L −→ K,{ , }(2,0)(1) : M × L −→ K, { , }(0)(2,1) : L×M −→ K,{ , } : M ×M −→ L

are called 3–dimensional Peiffer liftings. This data must satisfy the axioms (3CM1 −3CM18) given in [2].

Here we give the definition of 3–crossed complex of groups.

5.2. Definition. A 3–crossed complex C = {Cn, dn, { , }(2)(1), { , }} is a diagramof homomorphisms between groups

· · ·d5 // C4

d4 // C3d3 // C2

d2 // C1d1 // C0

such that dn−1dn = 1 for n ≥ 2 and ({ , }(2)(1), { , }, d3, d2, d1) is a 3–crossedmodule with π = cokerd1; hence ker d2 is a π–module. Moreover; Cn is a right π–module. A map f : C → C′ between 3–crossed complexes is a family of homomorphismsbetween groups for n ≥ 1

fn : Cn → C′n with fn−1dn = dnfn

such that (f4, f3, f2, f1) is a map between 3–crossed modules. Let X3Comp be thecategory of 3–crossed complexes and maps, we define the homotopy groups

π1(C) = π = coker(d1)

πn(C) =ker dn

imdn+1, n ≥ 2.

We call a 3–crossed complex C totally free if C1 is a free group, d2 : C1 → C0 isa free pre–crossed module, and d3, d2, d1 are given by a free 3–crossed module. LetFreeX3 ⊂ X3Comp be the full subcategory consisting of totally free 3–crossed com-plexes. In the second section, we introduced cofibrations in the category of 2–complexesby use of the universal properties of free extensions. In the same way we now definecofibrations in the category of 3–crossed complexes.

5.3. Definition. A map f : A→ B in 3–crossed complexes is a cofibration if f is a freeextension in each degree n, n ≥ 1.

Here we define a free extension in each degree n literally in the same way as in 2–crossed complexes.

The cofibrant objects in 3–crossed complexes are exactly the totally free 3–crossedcomplexes; hence we get the notation

FreeX3 = X3Compc

5.4. Lemma. Let An be an n–skeleton of a 3–crossed complex A, let fn−1 : An−1 →Bn−1 be a morphism in 3–crossed complex and let ∂n : Xn → Bn−1 be a function. Thena free extension f : An → Bn with basis ∂n exists provided that dn−1∂n = 1.

Proof. If X is a set, F (X) will denote the free group on X.For n = 1 we set free product of groups;

B1 = A1 ∗ F (X1).

38 H. Atik

For n = 2 we consider the free pre–crossed module

d2 : B2 = F ((A2 ∪X2)×B1)→ B1

with basis (f1d2, ∂2) : A2 ∪ X2 → B1 where A2 ∪ X2 is disjoint union. The inclusion

i : A2 → B2, however, is not a map between pre–crossed modules. Let U be the normalsubgroup of B2 generated by the relations

i(x)i(y)i(xy)−1 ' 1

i(xα)(f1αi(x))−1 ' 1

for x, y ∈ A2, α ∈ A1. Then d2 induces the pre–crossed module d2 : B2 = B2/U → B1.One readily checks that d2 has the universal property of free extensions.For n = 3, we consider the free pre–crossed module

d3 : B3 = F ((A3 ∪X3)×B2)→ B2

with basis (f2d3, ∂3) : A3 ∪X3 → B2 where A3 ∪X3 is disjoint union. The map j : A3 →B3, is not a map between pre–crossed modules. Let V be the normal subgroup of B3

generated by the relations

j(x)j(y)j(xy)−1 ' 1

j(xα)(f1αj(x))−1 ' 1

j{ , }(v)({ , }f2(v)× f2(v))−1 ' 1

where x, y ∈ A3, α ∈ A1, v ∈ A2×A2, and { , } : B2×B2 → B3. Then d3 induces the

pre–crossed module d3 : B3 = B3/V → B2.

For n = 4, the 3–crossed module B4 is as follows. Let

B4 → B3 → B2 → B1

be the free 3–crossed module with Peiffer map { , }(2)(1) : B3 × B3 → B4 with basis

(f3d4, ∂4) : A4 ∪X4 → B3. The inclusion k : A4 → B4, however, is not a map between

3–crossed modules. Let Y be the normal subgroup of B4 generated by the relations

k(x)k(y)k(xy)−1 ' 1

k(xα)(f1αk(x))−1 ' 1

k{ , }(2)(1)(p)({ , }(2)(1)f3(p)× f3(p))−1 ' 1

where x, y ∈ A4, α ∈ A1, p ∈ A3×A3. Then above diagram induces the commutativediagram

B3 ×B3

{,}(2)(1) //

{ , }(2)(1)��

B4

��{{vvvvvvvvv

B2 ×B2

{,} //

{{vvvvvvvvv

B3

{{vvvvvvvvv

uukkkkkkkk

kkkkkkkk

kkk

B4// B3

// B2// B1

where B4 = B4/Y is the quotient group with induced action of B1 and diagram is awell defined 3–crossed module. The bottom row with { , }(2)(1) is a well defined3–crossed module and one readily checks the universal property of free extensions issatisfied. Finally, for n ≥ 5, Bn is the direct sum of a free R–module generated by Xnand K. Hence K is the tensor product of An and group ring of π1(A) �

5.5. Theorem. The category of 3–crossed complexes with cofibrations and weak equiva-lences is a cofibration structure for which all objects are fibrant.


We use the same arguments as in the proof of parallel theorem in 2–crossed complexes.Moreover we prove that all objects are fibrant by showing that i : A → B is a strongdeformation retract morphism. A retraction r and a homotopy α : ir ' 1 is obtained bythe same formula. We construct push outs in X3Comp

Bf // B′

A

i

OO

// A′

i

OO

as follows. Let Bn be a free extension of A Then we set B′n as free extension of A′. Thebasis of B′ is given in degree n by the composition

fn−1d|Xn : Xn → An−1 → Bn−1, n ≥ 2.

As a result of above theorem and lemmas we have the following corollary.

5.6. Corollary. The category of totally free 3–crossed complexes is a cofibration category.

6. Homotopy of 3–crossed Modules

Let A = K∂3 // L

∂2 // M∂1 // N , A′ = K′

∂3 // L′∂2 // M ′

∂1 // N ′ betwo 3–crossed modules and f = (f0, f1, f2, f3) be a 3–crossed module morphism. Ahomotopy on f is a pair h = (h1, h2, h3) of maps where h1 : N →M ′, h2 : M → L′, andh3 : L→ K′satisfying equations below.

h1(nn′) = f0(n′)−1

h1(n)h1(n′)

h2(mm′) = ((f1m′)(h1∂1m

′))−1

({f1m′, f0∂1m′−1

(h1∂1m−1)}h2(m))h2(m′)

∂2h2(nm) = h1(n)f0(n)h1∂1(m−1)f0∂1(m−1)h1(n)f1(m−1)∂1h1(n)f1(n)h1∂1(m)

∂2h2(m)∂1h1(n−1)f0(n−1)

∂3h3(ll′) = h2∂2(ll′)−1f2(l′−1)h2∂2(l)∂3h3(l)f2(l′)h2∂2(l′)∂3h3(l′)

h3∂3(kk′) = f3(k′)−1h3∂3(k)f3(k′)h3∂3(k′)

∂3h3(nl) = h2∂2(nl)−1f0(n)f2(l−1)∂1h1(n)f2(l)h2∂2(l)∂3h3(l)∂1h1(n−1)f0(n−1)

h3∂3(nk) = f0(n)f3(k−1)∂1h1(n)f3(k)∂3h3(k)∂1h1(n−1)f0(n−1)

Such a function h is called a 2–quadratic f–derivation.

6.1. Proposition. Given a homotopy as above, the formulasf ′0(n) = f0(n)∂1h1(n)f ′1(m) = f1(m)h1∂1(m)∂2h2(m)f ′2(l) = f2(l)h2∂2(l)∂3h3(l)f ′3(k) = f3(k)h3∂3(k)

for all n ∈ N,m ∈M, l ∈ L define a morphism of 3–crossed modules.

Proof.

f ′0(nn′) = f0(nn′)∂1h1(nn′)

= f0(n)f0(n′)∂1(f0(n′)−1

h1(n)h1(n′))

= f0(n)f0(n′)f0(n′)−1∂1h1(n)f0(n′)∂1h1(n′))

= f0(n)∂1h1(n)f0(n′)∂1h1(n′))

= f ′0(n)f ′0(n′)

40 H. Atik

then f ′0 is a group homomorphism.

f ′1(mm′) = f1(mm′)h1∂1(mm′)∂2h2(mm′)

= f1(m)f1(m′)h1∂1(mm′)∂2h2(mm′)

= f1(m)f1(m′)f0∂1(m′)h1∂1(m)h1∂1(m′)∂2h2(mm′)

= f ′1(m)∂2h2(m−1)h1∂1(m−1)f1(m′)f0∂1(m′)h1∂1(m)h1∂1(m′)∂2h2(mm′).(1)

On the other hand

∂2h2(mm′) = h1∂1(m′)−1f1(m′)−1∂2({f1m′, f0∂1m′−1

(h1∂1m−1)}h2(m))f1(m′)h1∂1(m′)∂2h2(m′)

= h1∂1(m′)−1f1(m′)−1〈f1m′, f0∂1m′−1

(h1∂1m−1)〉∂2h2(m)f1(m′)h1∂1(m′)∂2h2(m′)

= h1∂1(m′)−1f1(m′)−1f1m′f0∂1m′−1

(h1∂1m−1)f1(m′)−1h1∂1(m)∂2h2(m)f ′1(m′)

= h1∂1(m′)−1f0∂1m′−1

(h1∂1m−1)f1(m′)−1h1∂1(m)∂2h2(m)f ′1(m′).(2)

From (1) and (2) we get

f ′1(mm′) = f ′1(m)f ′1(m′)

f ′2(ll′) = f2(ll′)h2∂2(ll′)∂3h3(ll′)

= f2(l)f2(l′)h2∂2(ll′)∂3h3(ll′)

= f ′2(l)∂3h3(l−1)h2∂2(l−1)f2(l′)h2∂2(ll′)∂3h3(ll′)

= f ′2(l)f ′2(l′)

f ′3(kk′) = f3(kk′)h3∂3(kk′)

= f3(k)f3(k′)h3∂3(kk′)

= f ′3(k)h3∂3(k−1)f3(k′)h3∂3(kk′)

= f ′3(k)f ′3(k′)

Then f = (f0, f1, f2, f3) is a homomorphism. Now we show that f is a morphism of3–crossed modules.

f ′1(nm) = f1(nm)h1∂1(nm)∂2h2(nm)

= f0(n)f1(m)h1(n∂1mn−1)∂2h2(nm)

= f0(n)f1(m)f0(n)−1f0(n)(f0∂1(m)−1

(h1(n))h1∂1(m))h1(n)−1∂2h2(nm)

= f0(n)f1(m)f0(n)−1f0(n)f0∂1(m)−1

(h1(n))h1∂1(m)f0(n)−1h1(n)−1∂2h2(nm)

= f ′0(n)f ′1(m)

f ′2(nl) = f2(nl)h2∂2(nl)∂3h3(nl)

= f0(n)f2(l)h2∂2(nl)∂3h3(nl)

= f0(n)f2(l)f0(n)−1h2∂2(nl)∂3h3(nl)

= f ′0(n)f ′2(l)


f ′3(nk) = f3(nk)h3∂3(nk)

= f0(n)f3(k)h3∂3(nk)

= f0(n)f3(k)f0(n)−1h3∂3(nk)

= f ′0(n)f ′3(k)

�

6.1. Homotopies of 3–crossed Complexes. Let A and A′ be two 3–crossed com-plexes and let f be a 3–crossed complex map A → A′. A 2–quadratic f–derivation isa sequence of maps hi : Ai → A′i+1 such that (h3, h2, h1) is a 2–quadratic f–derivationof 3–crossed modules and all the remaining maps are A1–equivariant for n = 4 andA1/∂A2–equivariant for n ≥ 5. We say that two 3–crossed complex maps are homotopicif there exists a 2–quadratic f–derivation such that

f ′1(a) = f1(a)∂2h1(a) and f ′n(a) = fn(a)(hn−1∂(a))(∂hn(a)) for n ≥ 2.

Here we can write a lemma by the light of the Lemma 3.6.

6.2. Lemma. Homotopy classes of category of totally free 3–crossed complexes is equiv-alent to the localization of 3–crossed complexes with respect to weak equivalences.

References

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Appl., 11, 161–187, 2009.

[3] Baues, H. J. Combinatorial homotopy and 4–dimensional complexes, (Walter de Gruyter,1991).

[4] Baues, H. J. Algebraic homotopy, (Cambridge Studies in Advanced Mathematics, 1998).

[5] Brown, R. and Higgins, P. J. The classifying space of a crossed complex, Math. Proc. Cam-bridge Phil. Soc., 110, 95–120, 1991.

[6] Brown R. and Higgins, P. J. Tensor Products and Homotopies for ω–groupoids and CrossedComplexes, J.P.A.A 47, 11–44, 1987.

[7] Brown, R. and Golanski, M. A model structure for the homotopy theory of crossed complexes,

Cah. Top. Geom. Diff. Cat, 30, 61–82, 1989.[8] Brown, R. and Icen, I. Homotopies and automorphisms of crossed modules over groupoids,

Appl. Categorical Structure, 11, 185–206, 2003.[9] Conduche, D. Modules croises ge neralises de longueur 2, Journal of Pure and Applied

Algebra, 34, 155–178, 1984.

[10] Kamps, K. H. Kan–Bedingungen und abstrakte Homotopie theorie, Math. Z., 124, 215–236,

1972.[11] Martin, Joao Faria Homotopies of 2–crossed complexes and the homotopy category of pointed

3–types, http://arxiv.org/pdf/math/0605364v1.pdf, 2011.[12] Mutlu, A. and Porter, T. Freeness conditions for 2–crossed modules and complexes, Theory

and Applications of Categories, 4 No. 8, 174–194, 1998.

[13] Quillen D. Lecture Notes in Math., Homotopical Algebra, 11, 185–206, 1967.[14] Radulescu–Banu, Andrei Cofibrations in Homotopy Theory, http://arxiv.org/abs/math/

0610009v4, 2009.

[15] Whitehead, J. H. C. Combinatorial homotopy II, Bull. Amer. Math. Soc., 55, 453–496, 1949.


Bounds for Resistance–Distance Spectral Radius


A. Dilek Gungor Maden ∗ 1, Ivan Gutman2, A. Sinan Cevik1

Abstract

Lower and upper bounds as well as Nordhauss-Gaddum-type results forthe resistance–distance spectral radius are obtained.

Keywords: Metric (in graph); Resistance Distance; Resistance-Distance Spectral Ra-dius

2000 AMS Classification: 15C12, 05C50.

1. Introduction and preliminaries

The resistance distance is a metric function on a graph, proposed by Klein and Randic[18]. The resistance distance Rij between the vertices vi and vj of a connected graph Gis defined to be equal to the resistance between the respective two nodes of an electricalnetwork, corresponding to G , in which the resistance between any two adjacent nodes is1 Ohm. It is known that the resistance distance satisfies the mathematical requirementsfor a distance ([2, 3, 17]).

It is known that the resistance distance can be expressed in terms of the eigenvaluesand eigenvectors of the Laplacian matrix and normalized Laplacian matrix associatedwith the network; for details on this matter see [8, 14, 15, 21, 22, 23, 24]. We also referto [4] for a new method for computing the resistance distances.

In [18] a molecular structure descriptor was introduced, equal to the sum of resistancedistances of all pairs of vertices of a molecular graph:

Kf = Kf(G) =∑i<j

Rij .

Eventually, it has been named the “Kirchhoff index” ([6]).The Kirchhoff index was much studied in mathematical chemistry. Details on its

theory can be found in the recent papers [5, 9, 12, 25, 27, 28, 32] and the references citedtherein.

∗Corresponding author : [email protected] University, Campus, 42075, Konya, Turkey2Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, SerbiaThe first and third authors are partially supported by TUBITAK and the Scientific Research

Project Office (BAP) of Selcuk University.The second author is supported by the Serbian Ministry of Science with Grant no. 144015G.

44 A. D. Gungor Maden, I. Gutman, A. S. Cevik

The Laplacian matrix of the graph G , denoted by L = ||Lij || , is a square matrix oforder n whose (i, j)-entry is defined by

Lij =

−1 if i 6= j and the vertices vi and vj are adjacent

0 if i 6= j and the vertices vi and vj are not adjacent

di if i = j

where di is the degree of the vertex vi . Further, J is the square matrix of order n whoseall elements are unity. Then for all connected graphs (with two or more vertices) the

matrix L+1

nJ is non-singular, its inverse

X = ‖xij‖ =

(L+

1

nJ

)−1

does exist and, by [22], Rij = xii +xjj−2xij . The resistance–distance matrix is an n×nmatrix R = R(G) = ‖Rij‖ . Note that the diagonal elements of R are equal to zero.

Since R is a real symmetric matrix, all its eigenvalues are real numbers. Let λ1(G) bethe maximum eigenvalue (i.e., the spectral radius) of R . Balaban et al. ([1]) proposedto use the maximum eigenvalues of distance–based matrices as structural descriptors inchemical researches. For more work along these line see [13, 16, 29, 31].

In this paper we present our results for the maximum eigenvalue of the resistance–distance matrix. We also provide some lower and upper bounds for λ1(G) for moleculargraphs and a few Nordhaus–Gaddum–type results [20]. (Recall that in [20] the boundshave been obtained for the sum of chromatic numbers of a graph and its complement.Eventually, such Nordhaus–Gaddum–type results were elaborated also for other graphinvariants.)

The following lemma is one of the key point in our considerations.

1.1. Lemma. [29, 30] Let B = (Bij) be an n × n nonnegative, irreducible, symmetricmatrix (n ≥ 2) with row sums B1, B2, . . . , Bn . If λ1(B) is the maximum eigenvalue ofB , then√∑n

i=1B2i

n≤ λ1(B) ≤ max

1≤j≤n

n∑i=1

Bij

√Bj

Bi

with equality holding if and only if B1 = B2 = · · · = Bn or if there is a permutationmatrix Q such that

QT BQ =

(0 CCT 0

),

where all the row sums of C are equal.

2. Bounds for λ1

2.1. Theorem. Let G be a connected graph with n ≥ 2 vertices. Then

(2.1)

√∑ni=1R

2i

n≤ λ1(G) ≤ max

1≤j≤n

n∑i=1

Rij

√Rj

Ri

where Ri is the sum of i-th row of the matrix R . Moreover equality holds in (2.1) if andonly if R1 = R2 = · · · = Rn .

Bounds for Resistance–Distance Spectral Radius 45

Proof. It is clear that the matrix R is irreducible for n ≥ 2 , and then, by Lemma 1.1, weobtain the inequality in (2.1). By definition, we know that Rij 6= 0 for i 6= j and Rij = 0otherwise. We note that for n ≥ 3 , there is no permutation matrix Q such that

QT RQ =

(0 CCT 0

)where all the row sums of C are equal. By Lemma 1.1, the equality in (2.1) holds if andonly if R1 = R2 = · · · = Rn . �

2.2. Corollary. Let G be a connected graph with n ≥ 2 vertices. Then

(2.2) λ1(G) ≥ 2Kf

n

with equality holding if and only if R1 = R2 = · · · = Rn or G ∼= Kn .

Proof. By the left part of the inequality given in (2.1) and in view of the Cauchy–Schwarzinequality, we obtain

λ1(G) ≥√∑n

i=1R2i

n≥∑n

i=1Ri

n=

2Kf

n

and equality holds if and only if R1 = R2 = · · · = Rn or G ∼= Kn . �

Note that trace[R] = 0 and denote by S = S(G) the trace of R2 . Therefore, theeigenvalues λi(G) , i = 1, 2, . . . , n , of R satisfy the relations

(2.3)

n∑i=1

λi(G) = 0

and

(2.4)

n∑i=1

λ2i (G) = S(G) .

It can be shown that S(G) is maximum for G ∼= Pn (where Pn is the n-vertex path),and S(G) is minimum for G ∼= Kn .

We first recall the long–time known fact ([19, 26]) that if G is a connected graph andG′ is the graph obtained from G by adding to it a new edge, then Kf(G) > Kf(G′) .This result is, of course, equivalent to the claim that if G is a connected graph and e isits edge, and if G− e is also connected, then Kf(G) < Kf(G− e) .

From this result it immediately follows that among connected n-vertex graphs, thecomplete graph has minimum Kirchhoff index.

It also follows that the (connected) graph with maximum Kirchhoff index must be atree. Because in the case of trees, the Kirchhoff and Wiener indices coincide, and becausewe know that among n-vertex trees, the path is the tree with maximum Wiener index([11, 10]), it follows that the path is also the n-vertex connected graph with maximumKirchhoff index.

Let G be the class of connected graphs whose resistance–distance matrices have exactlyone positive eigenvalue. In the following, we give upper and lower bounds for λ1(G) ofgraphs in the class G in terms of number of vertices and S(G) . Before that we state alemma that will be needed for determining the equality cases in the bounds given in thefollowing.


2.3. Lemma. [7] Let B be a nonnegative, irreducible, symmetric matrix with exactlytwo distinct eigenvalues. Then B = uuT + r I for some positive column vector u andsome r .

2.4. Theorem. Let G ∈ G with n ≥ 2 vertices. Then

(2.5) λ1(G) ≤√n− 1

nS(G)

with equality holding if and only if G ∼= Kn .

Proof. By (2.3), we have λ1(G) = −∑n

i=2 λi(G) . Further, by the Cauchy–Schwarzinequality and using (2.4),

λ21(G) =

[n∑

i=2

λi(G)

]2≤ (n− 1)

n∑i=2

nλ2i (G)

= (n− 1)[S(G)− λ2

1(G)]

with equality if and only if λ2(G) = · · · = λn(G) . We thus have

nλ21(G) ≤ (n− 1)S(G)

as required in (2.5).Suppose now that equality holds in (2.5). Then λ2(G) = · · · = λn(G) , and so the

matrix R has exactly two distinct eigenvalues. Now we prove that the diameter of G isone, i. e., G does not contain an induced shortest path Pm , m ≥ 3 .

Assume that G contains an induced shortest path Pm , m ≥ 3 . Let M be the principalsubmatrix of R indexed by the vertices in Pm . For an arbitrary matrix A , let θi(A)denote its i-th eigenvalue. Then, by the interlacing theorem,

θi(R) ≥ θi(M) ≥ θn−m+i(R), i = 1, 2, . . . ,m

or, in other words,

θ2(R) ≥ θ2(M) ≥ θ3(M) ≥ · · · ≥ θm(M) ≥ θn(R) .

This then shows that Pm has at most two distinct R-eigenvalues for m ≥ 3 , which isimpossible. Therefore G does not contain any two vertices at distance two or more, andhence it is complete. The other way around is quite obvious, i.e., if G ∼= Kn , then theequality holds in (2.5). �

2.5. Remark. By considering Lemma 2.3, the equality part of (2.5) in Theorem 2.4can be obtained quite similarly as in the proof of Theorem 3 of the paper [31].

The next result provides a lower bound for λ1(G) in terms of S(G) . Recall thatG is the class of connected graphs whose resistance–distance matrices have exactly onepositive eigenvalue.

2.6. Theorem. Let G ∈ G with n ≥ 2 . Then

(2.6) λ1(G) ≥√S(G)

2.

Equality holds in (2.6) if and only if G ∼= K2 .

Proof. We first note that λ1(G) > 0 and λ2(G) ≤ 0 . Then by (2.3),

2λ1(G) =

n∑i=1

|λi(G)| .


From (2.3) and (2.4) we also have

∑1≤i<j≤n

|λi(G)λj(G)| ≥

∣∣∣∣∣∣∑

1≤i<j≤n

λi(G)λj(G)

∣∣∣∣∣∣ =S(G)

2

and so

4λ21(G) =

[n∑

i=2

|λi(G)|

]2

=

n∑i=1

λ2i (G) + 2

∑i<j

|λi(G)λj(G)| ≥ 2S(G)

from which (2.6) follows.If we take n = 2 , then (2.6) is actually an equality. For the case n ≥ 3 , in order to

see that (2.6) is not an equality, the same approach can be applied as in [31, Theorem4]. Hence the result. �

3. Nordhaus-Gaddum-type bounds for λ1

In this section we consider general graphs, not only those from the class G . Weestablish some more bounds involving the Kirchhoff index Kf as well as Nordhaus–Gaddum–type bounds for λ1(G) .

Before that, consider some fundamental structural parameters of a connected (molec-

ular) graph G and its complement G . Let G be a connected (molecular) graph on n > 2vertices, m edges, maximum degree ∆ , second maximum degree ∆2 , minimum degreeδ , and second minimum degree δ2 . Further, assume that G has a connected complementG with m edges and related parameters ∆ , ∆2 , δ , and δ2 . As one can easily prove, thefollowing equalities exist between these parameters:

(3.1)

2(m+m) = n(n− 1)

∆ = n− 1− δ

δ = n− 1−∆

∆2 = n− 1− δ2

δ2 = n− 1−∆2

.

The following two lemmas have been recently proven in [9]. We nevertheless statethem here (without proof), since these are used for obtaining lower bounds for λ1(G)involving Kf (cf. Theorem 3.3 below).

3.1. Lemma. [9] Let G be a connected graph on n > 2 vertices and m edges withparameters ∆ , ∆2 , and δ as defined in (3.1). Then

(3.2) Kf(G) ≥ n

∆ + 1+

n

2m−∆− 1

[(n− 2)2 +

(∆2 − δ)2

∆2 δ

]with equality holding if and only if G ∼= K1,n−1 or G ∼= Kn .

3.2. Lemma. [9] Let G be a connected graph (not equal to Kn) on n > 2 vertices andm edges with parameters ∆ , ∆2 and δ as given in (3.1). Then

(3.3) Kf(G) ≥ 1 +n

δ+

n(n− 3)2

2m−∆− δ − 1

with equality holding if and only if G ∼= K1,n−1 .


After that we have the following lower bounds for λ1(G) . Their proofs are immediate,by considering inequalities in (2.2), (3.2), and (3.3).

3.3. Theorem. Let G be a connected graph on n > 2 vertices and m edges with param-eters ∆ , ∆2 and δ as given in (3.1). Then

(3.4) λ1(G) ≥ 2

{1

∆ + 1+

1

2m−∆− 1

[(n− 2)2 +

(∆2 − δ)2

∆2 δ

]}and

(3.5) λ1(G) ≥ 2

{1

n+n

δ+

n(n− 3)2

2m−∆− δ − 1

}.

Our next three theorems deal with Nordhaus-Gaddum type results for λ1(G) .

3.4. Theorem. Let G be a connected graph on n > 2 vertices and m edges, such thatits complement G is also connected. Then with the parameters given in (3.1),

λ1(G) + λ1(G) ≥ 2

{1

∆ + 1+

1

n− δ + (n− 2)2[

1

2m−∆− 1+

+1

n(n− 2)− 2m+ 1 + δ

]+

(1

2m−∆− 1

)((∆2 − δ)2

∆2 δ

)+

+

(1

n(n− 2)− 2m+ 1 + δ

)((∆− δ2)2

(n− 1− δ2)(n− 1−∆)

)}(3.6)

with equality holding if and only if G ∼= Kn .

Proof. Using the inequality (3.4) from Theorem 3.3, we arrive at

λ1(G) + λ1(G) ≥ 2

{1

∆ + 1+

1

2m−∆− 1

[(n− 2)2 +

(∆2 − δ)2

∆2 δ

]}

+ 2

{1

∆ + 1+

1

2m−∆− 1

[(n− 2)2 +

(∆2 − δ)2

∆2 δ

]}

= 2

{1

∆ + 1+

1

2m−∆− 1

[(n− 2)2 +

(∆2 − δ)2

∆2 δ

]}

+ 2

{1

n− δ +1

n(n− 2)− 2m+ 1 + δ

[(n− 2)2 +

(∆− δ2)2

(n− 1− δ2)(n− 1−∆)

]}and, by rearranging the terms in this final inequality, we obtain (3.6).

By using Corollary 2.2 and Lemma 3.1, one can easily see that the equality in (3.6)holds if and only if G ∼= Kn . Hence the result. �

3.5. Theorem. Let G be a connected graph on n > 2 vertices and m edges, such thatits complement G is also connected. Then with the parameters given in (3.1),

λ1(G) + λ1(G) ≥ 4

n+ 2

{(1

δ+

1

(n− 1−∆)

)

+ (n− 3)2[

1

2m−∆− δ − 1+

1

n(n− 3)− 2m+ 1 + ∆ + δ

]}.


Proof. Using the inequality (3.5) from Theorem 3.3 and applying similar arguments asin the proof of Theorem 3.4, we get the result. �

In view of the equality S(G) = n3 +S(G) and by using Theorem 2.4, we arrive at thefollowing result, which we present without proof.

3.6. Theorem. Let G ∈ G with n > 2 vertices, and let G be connected. Then,

λ1(G) + λ1(G) ≤√n− 1

n

[√S(G) +

√n3 + S(G)

]with equality holding if and only if G ∼= Kn .

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[1] Balaban, A. T., Ciubotariu, D. and Medeleanu, M. Topological indices and real numbervertex invariants based on graph eigenvalues or eigenvectors, J. Chem. Inf. Comput. Sci.

31, 517-523, 1991.

[2] Bapat R. B. Resistance distance in graphs, Math. Student 68, 87-98, 1999.[3] Bapat R. B. Resistance matrix of a weighted graph, MATCH Commun. Math. Comput.

Chem. 50, 73-82, 2004.

[4] Bapat R. B., Gutman, I. and Xiao, W. A simple method for computing resistance distance,Z. Naturforsch. 58a, 494-498, 2003.

[5] Bendito, E., Carmona, A., Encinas, A. M, Gesto, J. M. and Mitjana, M Kirchhoff indexes

of a network, Lin. Algebra Appl. 432, 2278-2292, 2010.[6] Bonchev, D., Balaban, A. T., Liu, X. and Klein, D. J. Molecular cyclicity and centricity of

polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances, Int. J.

Quantum Chem. 50, 1-20, 1994.[7] Cao, D., Chvatal, V, Hoffman, A, J. and Vince, A. Variations on a theorem of Ryser, Lin.

Algebra Appl. 260, 215-222, 1997.[8] Chen, H. and Zhang, F. Resistance distance and the normalized Laplacian spectrum, Discr.

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[9] Das, K. C., Gungor, A. D. and Cevik, A. S., On the Kirchoff index and the resistence distanceenergy of a graph, MATCH Commun. Math. Comput. Chem. 67(2), 541-556, 2012.

[10] Dobrynin, A. A., Entringer, R. and Gutman, I. Wiener index of trees: theory and applica-

tions, Acta Appl. Math. 66, 211-249, 2001.[11] Entringer, R. C., Jackson, D. E. and Snyder, D. A. Distance in graphs, Czech. Math. J. 26,

283-296, 1976.

[12] Guo, Q., Deng, H. and Chen, D. The extremal Kirchhoff index of a class of unicyclic graphs,MATCH Commun. Math. Comput. Chem. 61, 713-722, 2009.

[13] Gutman, I. and Medeleanu, M. On the structure–dependence of the largest eigenvalue of thedistance matrix of an alkane, Indian J. Chem. 37A, 569-573, 1998.

[14] Gutman, I. and Xiao, W. Generalized inverse of the Laplacian matrix and some applications,Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur. 129, 15-23, 2004.

[15] Gutman, I. and Xiao, W. Distance in trees and Laplacian matrix, Int. J. Chem. Model. 2,

327-334,2010.

[16] Ivanciuc, O., Ivanciuc, T. and Balaban, A. T. Quantitative structure–property relationshipevaluation of structural descriptors derived from the distance and reverse Wiener matrices,

Internet El. J. Mol. Des. 1, 467-487,2002.[17] Klein, D. J. Graph geometry, graph metrics, & Wiener, MATCH Commun. Math. Comput.

Chem. 35, 7-27, 1997.

[18] Klein, D. J. and Randic, M. Resistance distance, J. Math. Chem. 12, 81-95, 1993.

[19] Lukovits, I. Nikolic, S. and Trinajstic, N. Resistance distance in regular graphs, Int. J.Quantum Chem. 71, 217-225, 1999.

[20] Nordhaus, E. A. and Gaddum, J. W. On complementary graphs, Am. Math. Montly 63,175-177, 1956.


[21] Walikar, H. B., Misale, D. N., Patil, R. L. and Ramane, H. S. On the resistance distance of

a tree, El. Notes Discr. Math. 15, 244-245, 2003.

[22] Xiao, W. and Gutman, I. On resistance matrices, MATCH Commun. Math. Comput. Chem.49, 67-81, 2003.

[23] Xiao, W. and Gutman, I. Resistance distance and Laplacian spectrum, Theor. Chem. Acc.

110, 284-289, 2003.[24] Xiao, W. and Gutman, I. Relations between resistance and Laplacian matrices and their

applications, MATCH Commun. Math. Comput. Chem. 51, 119-127, 2004.

[25] Zhang, H., Jiang, X. and Yang, Y. Bicyclic graphs with extremal Kirchhoff index, MATCHCommun. Math. Comput. Chem. 61, 697-712, 2009.

[26] Zhang, H. and Yang, Y. Resistance distance and Kirchhoff index in circulant graphs, Int.

J. Quantum Chem. 107, 330-339, 2007.[27] Zhang, H., Yang, Y. and Li, C. Kirchhoff index of composite graphs, Discr. Appl. Math.

157, 2918-2927, 2009.[28] Zhang, W. and Deng, H. The second maximal and minimal Kirchhoff indices of unicyclic

graphs, MATCH Commun. Math. Comput. Chem. 61, 683-695, 2009.

[29] Zhou, B. On the spectral radius of nonnegative matrices, Australas. J. Comb. 22, 301-306,2000.

[30] Zhou, B. and Liu, B. On almost regular matrices, Util. Math. 54, 151-155, 1998.

[31] Zhou, B. and Trinajstic, N. On the largest eigenvalue of the distance matrix of a connectedgraph, Chem. Phys. Lett. 447, 384-387, 2007.

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46, 283-289, 2009.


SOFT RINGS RELATED TO

FUZZY SET THEORY

Xianping Liu∗, Dajing Xiang†, K. P. Shum‡ and Jianming Zhan § ¶


Abstract

We deal with soft rings based on some fuzzy sets, in particular, byusing the so called ∈ −soft sets and q-soft sets. Some characterizationtheorems of soft rings defined on soft sets are given and soft regularrings are hence characterized by special soft sets.

Keywords: Soft rings, Idealistic soft rings, Bi-idealistic soft rings, Quasi-idealistic softrings, (∈,∈ ∨q)-fuzzy subrings , (∈,∈ ∨ q)-fuzzy subrings , regular rings, soft regularrings.

2000 AMS Classification: 16Y60, 13E05, 03G25.

1. Introduction

In dealing with the complicated problems in economics, engineering and environmentalsciences, we are usually unable to apply the classical methods because there are variousuncertainties in these problems. There are three theories involved, namely, the theory ofprobability, the theory of fuzzy sets and the interval mathematics which are consideredas the fundamental tools in dealing with uncertainties, however all these theories havetheir own difficulties. Since uncertainties cannot be simply handled by using traditionalmathematics, one has to apply a wider range of existing theories such as probability,intuitionistic fuzzy sets, vague sets, interval mathematics, rough sets and so on to dealwith the situation. It is noted that all these theories have their own difficulties whichhave been pointed out in [13]. Maji et al. [12] and Molodtsov [13] have observed thatone reason for these difficulties may be due to the inadequacy of the parametrization

∗Hubei University for Nationalities, Department of Mathematics, Enshi, Hubei Province,445000, P. R. China.†Hubei University for Nationalities, Department of Mathematics, Enshi, Hubei Province,

445000, P. R. China.‡The University of Hong Kong, Department of Mathematics, Enshi, Pokfulam Road, Hong

Kong, China (SAR).§Hubei University for Nationalities, Department of Mathematics, Enshi, Hubei Province,

445000, P. R. China. E-mail: [email protected]¶Corresponding Author.

52 X. Liu, D. Xiang, K. P. Shum, J. Zhan

tools of the theory. In order to overcome these difficulties, Molodtsov [13] introduced theconcept of soft set which can be regarded as a new mathematical tool in dealing withuncertainties. Molodtsov also pointed out several directions for the applications of softsets. In recent years, research on soft set theory has been developed rapidly. Maji et al.[11] described the application of soft set theory to a decision making problem. Chen etal. [3] have recently presented a new definition of soft set parametrization reduction andcompared their definition to the related concept of attributes reduction in rough sets.The algebraic structure of set theories dealing with uncertainties has been investigatedby some authors and the algebraic theories dealing with uncertainties have also beenstudied by them. The most appropriate theories for dealing with uncertainties are basedon the theory of fuzzy sets established by Zadeh in 1965 (see [18, 19]).

The notion of soft sets for BCK/BCI-algebras was considered by Jun in [5]. Heintroduced the notions of soft BCK/BCI-algebras and investigated their basic properties[6]. Aktas et al. [1] further studied the basic concept of soft set theory and comparedsoft sets to fuzzy and rough sets, providing some examples to clarify their differences. Itis noteworthy that Feng et al. have started to investigate the structure of soft semiringsin [4].

After the concept of fuzzy sets introduced by Zadeh [8] in 1965, there are many pa-pers devoted to fuzzify the classical mathematics into fuzzy mathematics. Because theimportance of group theory in mathematics as well as its applications in many disci-plines, the notion of fuzzy subgroups was defined by Rosenfeld in 1971. Fuzzy algebraicstructures then play a prominent role in mathematics with a wider range of applicationsin many disciplines such as theoretical physics, computer sciences, control engineering,information sciences, coding theory, topological spaces and so on. These applicationsprovide sufficient motivation for researchers to review various concepts and results fromthe realm of abstract algebra to a broader framework of fuzzy setting. Some recentresearch on algebras can be found in [9, 10, 16, 17, 20-22].

The definition of soft rings has been recently proposed in [8]. Some properties of softrings were described and isomorphism theorems were established [8]. As a continuationof the above paper, we now continue to study the soft rings by using some special softsets. The concepts of idealistic soft rings, bi-idealistic soft rings and quasi-idealistic softrings generated by soft sets are introduced. As a consequence, the relationships betweensoft rings and their fuzzy subrings (ideals) are described. As a result, the regular ringsand soft regular rings are characterized by using special soft sets.

The notions, definitions and terminology used in this paper are standard. For somedefinitions and notations not given in this paper, the reader is referred to [8] and [18] ifnecessary.

2. Preliminaries

In this section, for the sake of completeness, we first cite some useful definitions andresults.

Throughout this paper, R is a ring.

2.1. Definition ([7]). A subring B of R is called a bi-ideal of R if BRB ⊂ B. A subringQ of R is called a quasi-ideal of R if RQ

⋂QR ⊂ Q.

It is clear that any left (right) ideal of R is a quasi-ideal of R, and any quasi-ideal ofR is a bi-ideal of R.

2.2. Definition ([10]). A fuzzy subset µ in a set X is a function µ : X −→ [0, 1]. If λand µ are two fuzzy subsets in X, then the intrinsic product λ ∗ µ is a fuzzy subset in X

Soft Rings Related to Fuzzy Set Theory 53

defined by(λ ∗ µ)(x) = sup

x=Σfiniteaibi

(min{λ(ai), µ(bi)}) .

2.3. Definition ([10]). A fuzzy subset µ in a set X of the form

µ(y) =

{t ∈ (0, 1] if y = x,

0 if y 6= x.

is called a fuzzy point with support x and value t, denoted by xt.

2.4. Definition ([10]). A fuzzy point xt is said to “belong to” (resp., be quasicoincidentwith) a fuzzy set µ, written by xt ∈ µ (resp., xtqµ) if µ(x) ≥ t (resp., µ(x) + t > 1).

If xt ∈ µ or xtqµ, then we write xt ∈ ∨qµ. If µ(x) < t (resp., µ(x) + t ≤ 1), then wewrite xt∈µ (resp., xtqµ). The symbol ∈ ∨q is to mean that ∈ ∨q does not hold.

2.5. Definition ([14]). A fuzzy set µ in a ring R is said to be a fuzzy subring of R ifthe following conditions hold for all x, y ∈ R : (1) µ(x − y) ≥ min{µ(x), µ(y)}, and (2)µ(xy) ≥ min{µ(x), µ(y)}.

2.6. Definition ([7, 14]). (i) A fuzzy set µ in a ring R is said to be a fuzzy left (right)ideal of R if the following conditions hold for all x, y ∈ R : (1) µ(x−y) ≥ min{µ(x), µ(y)},and (3) µ(xy) ≥ µ(y) (µ(xy) ≥ µ(x)).

(ii) A fuzzy set µ is said to be a fuzzy ideal of R if it is both a fuzzy left ideal of Rand a fuzzy right ideal of R.

(iii) A fuzzy set µ is called a fuzzy bi-ideal of R if it satisfies conditions (1), (2) and(4) µ(xyz) ≥ min{µ(x), µ(z)}, for all x, y, z ∈ R.

(iv) A fuzzy set µ is said to be a fuzzy quasi-ideal of R if the conditions (1) and (5)(µ ∗ χR)

⋂(χR ∗ µ)(x) ≤ µ(x) hold for all x ∈ R, where χR is the characteristic function

of R.

We next cite the following result.

2.7. Proposition ([7, 14]). A fuzzy set µ in a ring R is a fuzzy subring(ideal, bi-ideal,quasi-ideal) of R if and only if U(µ, α) = {x ∈ R | µ(x) ≥ α) is a subring(ideal, bi-ideal,quasi-ideal) of R, respectively.

3. Soft rings

The concept of soft set was first defined by Molodtsov in 1999 (see [13]).

3.1. Definition. (i) [13] Let P (U) be the power set of U and A ⊂ E, where E is a setof parameters. Then a pair (F,A) is called a soft set over U if F is a mapping given byF : A→ P (U).

(ii) [13] Let (F,A) and (G,B) be two soft sets over U . Then (F,A) is said to be a softsubset of (G,B) if the following conditions are satisfied:

(1) A ⊂ B and(2) for all x ∈ A, F (x) ⊂ G(x).

We now denote the above inclusion relationship by (F,A)⊂(G,B). Similarly, we call(F,A) a soft superset of (G,B) if (G,B) is a soft subset of (F,A). Denoted the aboverelationship by (F,A)⊃(G,B).

3.2. Definition. (i) [13] Two soft sets (F,A) and (G,B) over U are said to be soft equalif (F,A) is a soft subset of (G,B) and (G,B) is a soft subset of (F,A).

(ii) [1] The product of two soft sets (F,A) and (G,B) over U is the soft set (H,A×B),where H(x, y) = F (x)G(y), (x, y) ∈ A×B. This product is denoted by (F,A) ∗ (G,B) =(H,A×B).


(iii) [1] If (F,A) and (G,B) are soft sets over U , then we define the soft set (F,A) ∧(G,B), where (F,A) ∧ (G,B) is defined as (H,A × B), where H(x, y) = F (x) ∩ G(y),(x, y) ∈ A×B.

Let A be a nonempty set. We now use ρ to denote an arbitrary binary relation betweenan element of A and an element of the ring R. Then, a set-valued function F : A→ P (R)can be defined by F (x) = {y ∈ R | (x, y) ∈ ρ, x ∈ A}.

3.3. Definition. (i) [8] Let (F,A) be a soft set over R. Then (F,A) is said to be a softring over R if and only if F (x) is a subring of R for all x ∈ A. For the sake of convenience,the empty set ∅ here is regarded as a subring of R.

(ii) [8] Let (F,A) be a soft ring over a ring R. Then (F,A) is said to be an absolutesoft ring over R if F (x) = R for all x ∈ A.

3.4. Definition. Given a fuzzy set µ in any ring R and A ⊂ [0, 1], consider the followingtwo set-valued functions

F : A→ P(R), t 7→ {x ∈ R | xt ∈ µ}

and

Fq : A→ P(R), t 7→ {x ∈ R | xt qµ}.

Then (F , A) and (Fq, A) are called an “∈-soft set” and “q-soft set” over R, respec-tively.

In the following propositions, we characterize the soft rings over R by fuzzy subringsof R.

The following proposition follows directly from Proposition 2.7.

3.5. Proposition. Let µ be a fuzzy set in a ring R and A = [0, 1]. Then (F , A) is asoft ring over R if and only if µ is a fuzzy subring of R.

3.6. Proposition. Let µ be a fuzzy set in a ring R and A = [0, 1]. Then (Fq, A) is asoft ring over R if and only if µ is a fuzzy subring of R.

Proof. Assume that (Fq, A) is a soft ring over R. Then for all t ∈ A, Fq(t) is a subringof R. If there exist a, b ∈ R such that µ(a − b) < min{µ(a), µ(b)}, then we can chooset ∈ A such that µ(a − b) + t ≤ 1 < min{µ(a), µ(b)} + t. Hence, µ(a) + t > 1 andµ(b)+ t > 1, but µ(a− b)+ t ≤ 1, i.e., a, b ∈ Fq(t). However, we have a− b∈Fq(t), whichis a contradiction. Thus, µ(x− y) ≥ min{µ(x), µ(y)}, for all x, y ∈ R. In the same way,we can also prove that µ(xy) ≥ min{µ(x), µ(y)}, for all x, y ∈ R. Therefore, µ is a fuzzysubring of R.

Conversely, suppose that µ is a fuzzy subring of R. Let t ∈ A and x, y ∈ Fq(t). Thenµ(x − y) + t ≥ min{µ(x), µ(y)} + t > 1 and µ(xy) + t ≥min{µ(x), µ(y)} + t > 1, andso x − y ∈ Fq(t) and xy ∈ Fq(t). This proves that Fq(t) is a subring of R and hence(Fq, A) is a soft ring over R. �

We now introduce a special fuzzy subring of R.

3.7. Definition ([2]). We call a fuzzy set µ an (∈,∈ ∨q)-fuzzy subring of R if for allt, r ∈ (0, 1] and x, y ∈ R, the following conditions hold:

(A1) xt ∈ µ and yr ∈ µ imply (x− y)min(t,r) ∈ ∨qµ,(A2) xt ∈ µ and yr ∈ µ imply (xy)min(t,r) ∈ ∨qµ.

In view of the above definition, we have the following lemma.


3.8. Lemma ([2]). A fuzzy set µ in a ring R is an (∈,∈ ∨q)-fuzzy subring of R if andonly if for all x, y ∈ R, the following conditions hold:

(B1) µ(x− y) ≥ min{µ(x), µ(y), 0.5},(B2) µ(xy) ≥ min{µ(x), µ(y), 0.5}

In the following theorem, we show that the soft rings can be described by the (∈,∈ ∨q)-fuzzy subrings of R.

3.9. Theorem. Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (F , A) is a softring over R if and only if µ is an (∈,∈ ∨q)-fuzzy subring of R.

Proof. Assume that (F , A) is a soft ring over R. Then F (t) is a subring of R for allt ∈ A. If there exist x, y ∈ R such that µ(x − y) < min{µ(x), µ(y), 0.5}, then we canchoose t ∈ (0, 1] such that µ(x − y) < t ≤ min{µ(x), µ(y), 0.5}. Thus 0 < t ≤ 0.5,µ(x) ≥ t, µ(y) ≥ t and µ(x − y) < t, that is, x, y ∈ F (t), but x − y∈F (t) which isa contradiction. Hence, µ(x − y) ≥ min{µ(x), µ(y), 0.5}. In the same way, we can alsoprove that µ(xy) ≥ min{µ(x), µ(y), 0.5}. By Lemma 3.8, µ is an (∈,∈ ∨q)-fuzzy subringof R.

Conversely, suppose that µ is an (∈,∈ ∨q)-fuzzy subring of R. Let t ∈ A. Then,by Lemma 3.8, we can deduce that µ(x − y) ≥ min{µ(x), µ(y), 0.5} and µ(xy) ≥min{µ(x), µ(y), 0.5}, for all x, y ∈ R. If x, y ∈ F (t), then µ(x) ≥ t and µ(y) ≥ t.These imply that µ(x− y) ≥ min{µ(x), µ(y), 0.5} ≥ min{t, 0.5} = t and so x− y ∈ F (t).We can also show that xy ∈ F (t). Thus F (t) is a subring of R and (F , A) is indeed asoft ring over R. �

The (∈,∈ ∨ q)-fuzzy subring of R can be defined as the same as an (∈,∈ ∨ q)-fuzzyh-bi-ideals of R in [10].

3.10. Definition. A fuzzy set µ is said to be an (∈,∈ ∨ q)-fuzzy subring of R if for allt, r ∈ (0, 1] and x, y ∈ R, the following conditions hold:

(C1) (x− y)min(t,r)∈µ implies xt∈ ∨ qµ or yr∈ ∨ qµ(C2) (xy)min(t,r)∈µ implies xt∈ ∨ qµ or yr∈ ∨ qµ.

We have the following same conclusion as in [10].

3.11. Lemma. A fuzzy set µ in a ring R is an (∈,∈∨ q)-fuzzy subring of R if and onlyif for all x, y ∈ R, the following conditions hold:

(D1) max{µ(x− y), 0.5} ≥ min{µ(x), µ(y)},(D2) max{µ(xy), 0.5} ≥ min{µ(x), µ(y)}.

3.12. Theorem. Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (F , A) is asoft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy subring of R.

Proof. Let (F , A) be a soft ring over R. Then F (t) is a subring of R for all t ∈ A. Ifthere exist x, y ∈ R such that max{µ(x − y), 0.5} < t ≤ min{µ(x), µ(y)}, then t ∈ A,x, y ∈ F (t), but x − y∈F (t), which is a contradiction. Hence, max{µ(x − y), 0.5} ≥min{µ(x), µ(y)}. In the same way, we can prove that max{µ(xy), 0.5} ≥ min{µ(x), µ(y)}.Hence, by Lemma 3.11, µ is an (∈,∈ ∨ q)-fuzzy subring of R.

Conversely, suppose that µ is an (∈,∈∨q)-fuzzy subring ofR. Then x, y ∈ R,max{µ(x−y), 0.5} ≥ min{µ(x), µ(y)} and max{µ(xy), 0.5} ≥ min{µ(x), µ(y)}. If we let t ∈ A withx, y ∈ F (t), then µ(x) ≥ t > 0.5, µ(y) ≥ t > 0.5, and hence max{µ(x − y), 0.5} ≥min{µ(x), µ(y)} ≥ t,max{µ(xy), 0.5} ≥ min{µ(x), µ(y)} ≥ t. Thus, µ(x − y) ≥ t andµ(xy) ≥ t, that is, x− y ∈ F (t) and xy ∈ F (t). These show that F (t) is a subring of Rand (F , A) is a soft ring over R. �

We next formulate the following theorems by using q-soft sets.


3.13. Theorem. Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (Fq, A) is asoft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy subring of R.

Proof. Assume that (Fq, A) is a soft ring over R. Then Fq(t) is a subring of R for allt ∈ A. If there exist x, y ∈ R such that max{µ(x − y), 0.5} < min{µ(x), µ(y)}, then wecan select t ∈ A such that max{µ(x−y), 0.5}+ t ≤ 1 < min{µ(x), µ(y)}+ t, x, y ∈ Fq(t),but x−y∈Fq(t), which is a contradiction. Hence, max{µ(x−y), 0.5} ≥ min{µ(x), µ(y)}.In the same way, we can also prove that max{µ(xy), 0.5} ≥ min{µ(x), µ(y)}. It hencefollows from Lemma 3.11 that µ is an (∈,∈ ∨ q)-fuzzy subring of R.

Conversely, suppose that µ is an (∈,∈∨q)-fuzzy subring ofR. Then x, y ∈ R,max{µ(x−y), 0.5} ≥ min{µ(x), µ(y)} and max{µ(xy), 0.5} ≥ min{µ(x), µ(y)}. Let t ∈ A so thatx, y ∈ Fq(t). Then µ(x) + t > 1, µ(y) + t > 1, hence, max{µ(x − y), 0.5} + t ≥min{µ(x), µ(y)}+ t, max{µ(xy), 0.5}+ t ≥ min{µ(x), µ(y)}+ t. Thus, µ(x− y) + t > 1and µ(xy) + t > 1, i.e., x − y ∈ Fq(t) and xy ∈ Fq(t), and so Fq(t) is a subring of Rand (Fq, A) is a soft ring over R. �

3.14. Theorem. Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (Fq, A) is asoft ring over R if and only if µ is an (∈,∈ ∨q)-fuzzy subring of R.

Proof. Let (Fq, A) be a soft ring over R. Then Fq(t) is a subring of R, for all t ∈ A.If there exist x, y ∈ R such that µ(x − y) < min{µ(x), µ(y), 0.5}, then we can chooset ∈ (0.5, 1] such that µ(x− y) + t ≤ 1 < min{µ(x), µ(y), 0.5}+ t. Thus x, y ∈ Fq(t), butx− y∈Fq(t). This is a contradiction. Hence, µ(x− y) ≥ min{µ(x), µ(y), 0.5}. By usingthe same arguments, we can prove that µ(xy) ≥ min{µ(x), µ(y), 0.5}. It follows fromLemma 3.8 that µ is an (∈,∈ ∨q)-fuzzy subring of R.

Conversely, suppose that µ is an (∈,∈ ∨q)-fuzzy subring of R. Using Lemma 3.8, wehave µ(x − y) ≥ min{µ(x), µ(y), 0.5} and µ(xy) ≥ min{µ(x), µ(y), 0.5} for all x, y ∈ R.Let t ∈ A, x, y ∈ Fq(t), then µ(x) + t > 1 and µ(y) + t > 1. These imply thatµ(x− y) + t ≥ min{µ(x), µ(y), 0.5}+ t > 1, and so x− y ∈ Fq(t). We can also similarlyprove that xy ∈ Fq(t). Thus Fq(t) is a subring of R and (Fq, A) is a soft ring overR. �

Same as the definition in [10], we give the following definition.

3.15. Definition. Let α, β ∈ (0, 1] with α < β. Then a fuzzy set µ is called an (α, β)-fuzzy subring of R if the following conditions are satisfied for any x, y ∈ R:

(E1) max{µ(x− y), α} ≥ min{µ(x), µ(y), β},(E2) max{µ(xy), α} ≥ min{µ(x), µ(y), β}.

3.16. Theorem. Let µ be a fuzzy set in a ring R with A = (α, β]. Then (F , A) is a softring over R if and only if µ is an (α, β)-fuzzy subring of R.

Proof. Assume that (F , A) is a soft ring over R. Then F (t) is a subring of R for allt ∈ A. If there exist x, y ∈ R such that max{µ(x − y), α} < min{µ(x), µ(y), β}, thenwe can select t ∈ (α, β] such that max{µ(x − y), α} < t ≤ min{µ(x), µ(y), β}. Thusµ(x) ≥ t and µ(y) ≥ t, but µ(x− y) < t, that is , x, y ∈ F (t), but x− y∈F (t), which isa contradiction. Hence, max{µ(x− y), α} ≥ min{µ(x), µ(y), β}. Similarly, we can provethat max{µ(xy), α} ≥ min{µ(x), µ(y), β}. Consequently, µ is an (α, β)-fuzzy subring ofR.

Conversely, suppose that µ is an (α, β)-fuzzy subring of R. For any t ∈ A, ifx, y ∈ F (t), then µ(x) ≥ t and µ(y) ≥ t. These imply that max{µ(x − y), α} ≥min{µ(x), µ(y), β} ≥ t and max{µ(xy), α} ≥ min{µ(x), µ(y), β} ≥ t. Thus, µ(x− y) ≥ tand µ(xy) ≥ t, that is, x− y ∈ F (t) and xy ∈ F (t), and so F (t) is a subring of R and(F , A) is indeed a soft ring over R. �


4. Idealistic soft rings, bi-idealistic soft rings and quasi-idealisticsoft rings

We divide this section into three parts. In Subsection 4.1, we describe the idealisticsoft rings. In Subsection 4.2, we describe the bi-idealistic soft rings. In Subsection 4.3,we consider the quasi-idealistic soft rings.

4.1. Idealistic soft rings.

4.1. Definition. Let (F,A) be a soft set over a ring R. Then (F,A) is said to be a left(right) idealistic soft ring over R if and only if F (x) is a left (right) ideal of R, for allx ∈ A. We now call (F,A) an idealistic soft ring over R if and only if (F,A) is both aright idealistic soft ring over R and a left idealistic soft ring over R.

For the sake of convenience, we now regard the empty set ∅ here as an ideal of R.

4.2. Example. Let Z6 = {0, 1, 2, 3, 4, 5} and (F,A) be a soft set over Z6, where A ={2, 3, 4, 5} and F : A → P (Z6) is defined by F (x) = {y ∈ Z6 | xρy ⇐⇒ xy = 0}, forall x ∈ A. Then it is clear that F (2) = {0, 3}, F (3) = {0, 2, 4}, F (4) = {0, 3} andF (5) = {0} are ideals of Z6. Clearly, (F,A) is an idealistic soft ring over Z6.

The proofs of the following propositions are easy (refer to Proposition 2.7 and Propo-sition 3.6, respectively).

4.3. Proposition. Let µ be a fuzzy set in a ring R and A = [0, 1]. Then (F , A) is aleft (right) idealistic soft ring over R if and only if µ is a fuzzy left (right) ideal of R.

4.4. Proposition. Let µ be a fuzzy set in a ring R and A = [0, 1]. Then (Fq, A) is aleft (right) idealistic soft ring over R if and only if µ is a fuzzy left (right) ideal of R.

We now consider the following special fuzzy left (right) ideals of R.

4.5. Definition. A fuzzy set µ is called an (∈,∈ ∨q)-fuzzy left (right) ideal of R if forall t, r ∈ (0, 1] and x, y ∈ R, the following conditions are satisfied:

(F1) xt ∈ µ and yr ∈ µ imply (x− y)min(t,r) ∈ ∨qµ,(F2) yt ∈ µ (xt ∈ µ) imply (xy)t ∈ ∨qµ.

By the above definition, we have the following lemma.

4.6. Lemma. A fuzzy set µ in a ring R is an (∈,∈ ∨q)-fuzzy left (right) ideal of R ifand only if for x, y ∈ R, the following conditions are satisfied:

(G1) µ(x− y) ≥ min{µ(x), µ(y), 0.5},(G2) µ(xy) ≥ min{µ(y), 0.5} (µ(xy) ≥ min{µ(x), 0.5}).

Proof. In view of Definition 4.5, we need to prove that conditions (F1) and (F2) areequivalent to conditions (G1) and (G2). Clearly, (F1)⇐⇒ (G1) by Lemma 3.8. We onlyprove that (F2)⇐⇒ (G2).

To prove that (F2) =⇒ (G2): Assume that there exist x, y ∈ R with µ(xy) < t ≤min{µ(y), 0.5} (µ(xy) < t ≤ min{µ(x), 0.5}). Then 0 < t ≤ 0.5 and yt ∈ µ (xt ∈ µ), butxytµ. Since µ(xy) + t ≤ 1, (xy)tqµ. It follows that (xy)t∈ ∨qµ, which is a contradiction.Hence (G2) holds.

To prove that (G2) =⇒ (F2): Let yt ∈ µ (xt ∈ µ). Then µ(y) ≥ t (µ(x) ≥ t). Nowµ(xy) ≥ min {µ(y), 0.5} ≥ min{t, 0.5}(µ(xy) ≥ min{µ(x), 0.5} ≥ min{t, 0.5}). If t > 0.5,then µ(xy) ≥ 0.5. This implies that µ(xy) + t > 1. If t ≤ 0.5, then µ(xy) ≥ t. Therefore,(xy)t ∈ ∨qµ. �

The proof of the following theorem is similar to the proof of Theorem 3.9 and is henceomitted.


4.7. Theorem. Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (F , A) is a left(right) idealistic soft ring over R if and only if µ is an (∈,∈ ∨q)-fuzzy left (right) idealof R.

Same as in [10], we give the following definition.

4.8. Definition. A fuzzy set µ is said to be an (∈,∈ ∨ q)-fuzzy left (right) ideal of R iffor all t, r ∈ (0, 1] and x, y ∈ R, the following conditions hold :

(H1) (x− y)min(t,r)∈µ implies xt∈ ∨ qµ or yr∈ ∨ qµ,(H2) (xy)t∈µ implies yt∈ ∨ qµ (xt∈ ∨ qµ).

The following lemma describes the properties of (∈,∈ ∨ q)-fuzzy left (right) ideals ofR.

4.9. Lemma. A fuzzy set µ in a ring R is an (∈,∈ ∨ q)-fuzzy left (right) ideal of R ifand only if the following conditions hold for all x, y ∈ R:

(I1) max{µ(x− y), 0.5} ≥ min{µ(x), µ(y)},(I2) max{µ(xy), 0.5} ≥ µ(y) (max{µ(xy), 0.5} ≥ µ(x)).

Proof. It is known that (H1)⇐⇒ (I1), we only need to prove (H2)⇐⇒ (I2).To prove (H2) =⇒ (I2): If there exist x, y ∈ R such that max{µ(xy), 0.5} < µ(y)

(max{µ(xy), 0.5} < µ(x)), then we can select t ∈ (0, 1] such that max{µ(xy), 0.5} < t ≤µ(y)(max{µ(xy), 0.5} < t ≤ µ(x)), and so 0.5 < t ≤ 1 and yt ∈ µ (xt ∈ µ), but (xy)t∈µ.By H(2), we have ytqµ (xtqµ). This implies µ(y) + t ≤ 1 (µ(x) + t ≤ 1), a contradiction.

To prove (I2) =⇒ (H2): Let t ∈ (0, 1] and (xy)t∈µ. Then µ(xy) < t.(a) If µ(xy) ≥ µ(y) (µ(xy) ≥ µ(x)), then µ(y) < t (µ(x) < t). It follows that yt∈µ

(xt∈µ). Thus, yt∈ ∨ qµ (xt∈ ∨ qµ).(b) If µ(xy) < µ(y) (µ(xy) < µ(x)), then by (I2), 0.5 ≥ µ(y)(0.5 ≥ µ(x)). Now, if

for µ(y) < t (µ(x) < t), then yt∈µ (xt∈µ) and if µ(y) ≥ t (µ(x) ≥ t), then µ(y) + t ≤ 1(µ(x) + t ≤ 1). It follows that ytqµ (xtqµ). Thus, yt∈ ∨ qµ (xt∈ ∨ qµ). �

4.10. Theorem. Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (F , A) is aleft (right) idealistic soft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy left (right)ideal of R.

Proof. The proof is similar to Theorem 3.12 and is hence omitted. �

We now characterize the left (right) idealistic soft rings over R by using q-soft sets.

4.11. Theorem. Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (Fq, A) is aleft (right) idealistic soft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy left (right)ideal of R.

Proof. Let (Fq, A) be a left (right) idealistic soft ring over R. Then Fq(t) is a left(right) ideal of R for every t ∈ A. If there exist x, y ∈ R such that max{µ(x− y), 0.5} <min{µ(x), µ(y)}, then we can select t ∈ A such that max{µ(x − y), 0.5} + t ≤ 1 <min{µ(x), µ(y)} + t and x, y ∈ Fq(t), but x − y∈Fq(t), this is a contradiction. Hence,max{µ(x−y), 0.5} ≥ min{µ(x), µ(y)}. If there exist c, d ∈ R such that max{µ(cd), 0.5} <µ(d)(max{µ(cd), 0.5} < µ(c)}), then we can select t ∈ A such that max{µ(cd), 0.5} +t ≤ 1 < µ(d) + t(max{µ(cd), 0.5} + t ≤ 1 < µ(c) + t). This leads to d ∈ Fq(t),c ∈ R (c ∈ Fq(t), d ∈ R), but cd∈Fq(t), a contradiction. Hence, max{µ(xy), 0.5} ≥µ(y)(max{µ(xy), 0.5} ≥ µ(x)) for all x, y ∈ R. It follows that µ is a left (right) (∈,∈∨q)-fuzzy ideal of R.

Conversely, suppose that µ is an (∈,∈ ∨ q)-fuzzy left (right) ideal of R. Then x, y ∈R,max{µ(x−y), 0.5} ≥ min{µ(x), µ(y)} and max{µ(xy), 0.5} ≥ µ(y)(max{µ(xy), 0.5} ≥µ(x)). For t ∈ A, if x, y ∈ Fq(t), then µ(x) + t > 1 and µ(y) + t > 1. These lead to


max{µ(x − y), 0.5} + t ≥ min{µ(x), µ(y)} + t > 1, i.e., x − y ∈ Fq(t). Let x ∈ Fq(t),z ∈ R. Then max{µ(zx), 0.5} + t ≥ µ(x) + t > 1(max{µ(xz), 0.5} + t ≥ µ(x) + t > 1).Thus, µ(zx) + t > 1(µ(xz) + t > 1), i.e., zx ∈ Fq(t) (xz ∈ Fq(t)), and hence Fq(t) is aleft (right) ideal of R and (Fq, A) is a left (right) idealistic soft ring over R. �

4.12. Theorem. Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (Fq, A) is aleft (right) idealistic soft ring over R if and only if µ is a (∈,∈ ∨q)-fuzzy left (right) idealof R.

Proof. Assume that (Fq, A) is a left (right) idealistic soft ring over R. Then Fq(α)is a left (right) ideal of R, for all α ∈ A. If there exist x, y ∈ R such that µ(x −y) < min{µ(x), µ(y), 0.5}, then we can select t ∈ A such that µ(x − y) + t ≤ 1 <min{µ(x), µ(y), 0.5}+t. Thus µ(x)+t > 1, µ(y)+t > 1, µ(x−y)+t ≤ 1. i.e., x, y ∈ Fq(t),but x−y∈Fq(t). This is a contradiction. Hence we have µ(x−y) ≥ min{µ(x), µ(y), 0.5}.If there exist a, b ∈ R such that µ(ab) < min{µ(b), 0.5} (µ(ab) < min{µ(a), 0.5}), thenwe can select t ∈ A such that µ(ab) + t ≤ 1 < min{µ(b), 0.5} + t (µ(ab) + t ≤ 1 <min{µ(a), 0.5} + t), then b ∈ Fq(t), a ∈ R (a ∈ Fq(t), b ∈ R) , but ab∈Fq(t), whichis absurd. Hence, we have µ(xy) ≥ min{µ(y), 0.5} (µ(xy) ≥ min{µ(x), 0.5}), and thisproves that µ is an (∈,∈ ∨q)-fuzzy left (right)ideal of R.

Conversely, let µ be an (∈,∈ ∨q)-fuzzy left (right) ideal of R. If t ∈ A, then, byLemma 4.6, we can get µ(x − y) ≥ min{µ(x), µ(y), 0.5} and µ(xy) ≥ min{µ(y), 0.5}(µ(xy) ≥ min{µ(x), 0.5}) for all x, y ∈ R. If x, y ∈ Fq(t), then µ(y) + t > 1 andµ(x) + t > 1. These imply that µ(x − y) + t ≥ min{µ(x), µ(y), 0.5} + t > 1, andso x − y ∈ Fq(t). If x ∈ Fq(t) and z ∈ R, then Fq(x) + t > 1. This leads toµ(zx) + t ≥ min{µ(x), 0.5} + t > 1 (µ(xz) + t ≥ min{µ(x), 0.5} + t > 1). Hence,zx ∈ Fq(t) (xz ∈ Fq(t)). Thus Fq(t) is a left (right) ideal of R and (Fq, A) is a left(right) idealistic soft ring over R. �

We now introduce the concept of (α, β)-fuzzy left (right) ideals of R.

4.13. Definition. Let α, β ∈ (0, 1] with α < β. Then a fuzzy set µ is called an (α, β)-fuzzy left (right) ideal of R if for x, y ∈ R, the following conditions are satisfied:

(J1) max{µ(x− y), α} ≥ min{µ(x), µ(y), β},(J2) max{µ(xy), α} ≥ min{µ(y), β}(max{µ(xy), α} ≥ min{µ(x), β}).

The following proposition follows from Theorem 3.16.

4.14. Theorem. Let µ be a fuzzy set in a ring R and A = (α, β]. Then (F , A) is a left(right) idealistic soft ring over R if and only if µ is an (α, β)-fuzzy left (right) ideal of R.

4.2. Bi-idealistic soft rings.

4.15. Definition. Let (F,A) be a soft set over a ring R. Then (F,A) is said to be abi-idealistic soft ring over R if and only if F (x) is a bi-ideal of R for all x ∈ A. For thesake of convenience, we now regard the empty set ∅ as a bi-ideal of R.

4.16. Example. In Example 4.2, (F,A) is a bi-idealistic soft ring over Z6.

The proofs of following propositions are straightforward and are omitted.

4.17. Proposition. Let µ be a fuzzy set in a ring R and A = [0, 1]. Then (F , A) is abi-idealistic soft ring over R if and only if µ is a fuzzy bi-ideal of R.

4.18. Proposition. Let µ be a fuzzy set in a ring R and A = [0, 1]. Then (Fq, A) is abi-idealistic soft ring over R if and only if µ is a fuzzy bi-ideal of R.

As the same as [10], we also have the following definitions and lemmas.


4.19. Definition. A fuzzy set µ is said to be an (∈,∈ ∨q)-fuzzy bi-ideal of R if for allt, r ∈ (0, 1] and x, y, z ∈ R, the following conditions hold:

(K1) xt ∈ µ and yr ∈ µ imply (x− y)min(t,r) ∈ ∨qµ,(K2) xt ∈ µ and yr ∈ µ imply (xy)min(t,r) ∈ ∨qµ,(K3) xt ∈ µ and zr ∈ µ imply (xyz)min(t,r) ∈ ∨qµ.

4.20. Lemma. A fuzzy set µ in a ring R is an (∈,∈ ∨q)-fuzzy bi-ideal of R if and onlyif the following conditions hold for any x, y, z ∈ R:

(L1) µ(x− y) ≥ min{µ(x), µ(y), 0.5},(L2) µ(xy) ≥ min{µ(x), µ(y), 0.5},(L3) µ(xyz) ≥ min{µ(x), µ(z), 0.5}.

4.21. Definition. A fuzzy set µ is said to be an (∈,∈ ∨ q)-fuzzy bi-ideal of R if for allt, r ∈ (0, 1], the following conditions hold for x, y ∈ R:

(M1) (x− y)min(t,r)∈µ implies xt∈ ∨ qµ or yr∈ ∨ qµ,(M2) (xy)min(t,r)∈µ implies xt∈ ∨ qµ or yr∈ ∨ qµ,(M3) (xyz)min(t,r)∈µ implies xt∈ ∨ qµ or zr∈ ∨ qµ.

It is easy to see that the (∈,∈ ∨ q)-fuzzy bi-ideal of R has the following properties:

4.22. Lemma. A fuzzy set µ in R is an (∈,∈ ∨ q)-fuzzy bi-ideal of R if and only if forall x, y ∈ R, the following conditions hold:

(N1) max{µ(x− y), 0.5} ≥ min{µ(x), µ(y)},(N2) max{µ(xy), 0.5} ≥ min{µ(x), µ(y)}.(N3) max{µ(xyz), 0.5} ≥ min{µ(x), µ(z)}.

4.23. Definition. Let α, β ∈ (0, 1] with α < β. Then a fuzzy set µ is called an (α, β)-fuzzy bi-ideal of R if x, y, z ∈ R, the following conditions hold:

(O1) max{µ(x− y), α} ≥ min{µ(x), µ(y), β},(O2) max{µ(xy), α} ≥ min{µ(x), µ(y), β},(O3) max{µ(xyz), α} ≥ min{µ(x), µ(z), β}.

In the following theorem, the properties of the bi-idealistic soft rings will be described.The proofs are similar to Theorem 3.9, Theorem 3.12, Theorem 3.13, Theorem 3.14 andTheorem 3.16, respectively.

4.24. Theorem. (i) Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (F , A) isa bi-idealistic soft ring over R if and only if µ is an (∈,∈ ∨q)-fuzzy bi-ideal of R.

(ii) Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (F , A) is a bi-idealisticsoft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy bi-ideal of R.

(iii) Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (Fq, A) is a bi-idealisticsoft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy bi-ideal of R.

(iv) Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (Fq, A) is a bi-idealisticsoft ring over R if and only if µ is an (∈,∈ ∨q)-fuzzy bi-deal of R.

(v) Let µ be a fuzzy set in a ring R and A = (α, β]. Then (F , A) is a bi-idealistic softring over R if and only if µ is an (α, β)-fuzzy bi-ideal of R.

4.3. Quasi-idealistic soft rings.

4.25. Definition. Let (F,A) be a soft set over a ring R. Then (F,A) is said to be aquasi-idealistic soft ring over R if and only if F (x) is a quasi-ideal of R for all x ∈ A.

For the sake of convenience, we now regard the empty set ∅ here as a quasi-ideal of R.


4.26. Example. Let R = M2(R), A =

{(a 00 0

)|a 6= 0

}and F (x) = {y ∈ R | xy = 0}.

Then (F,A) is a soft ring over R. ∀x ∈ A, F (x) =

{(0 0x1 x2

)|x1, x2 ∈ R

}. Let(

y1 y2

y3 y4

)∈ R. Then(

y1 y2

y3 y4

)(0 0x1 x2

)=

(x1y2 x2y2

x1y4 x2y4

)and(

0 0x1 x2

)(y1 y2

y3 y4

)=

(0 0

x1y1 + x2y3 x1y2 + x2y4

). Then

F (x)R ∩RF (x) =

{(0 0m n

)|m,n ∈ R

}. Because(

a 00 0

)(0 0m n

)=

(0 00 0

)and

F (x)R ∩RF (x) ⊂ F (x), (F,A) is a quasi-idealistic soft ring over R.

Since any left (right) ideal of a ring R is a quasi-ideal of R and any quasi-ideal of R isa bi-ideal of R, by Proposition 2.7 and Proposition 3.6 we can easily deduce the followingproposition.

4.27. Proposition. (i) Any left (right) idealistic soft ring over R is a quasi-idealisticsoft ring over R.

(ii) Any quasi-idealistic soft ring over R is a bi-idealistic soft ring over R.(iii) Let µ be a fuzzy set in a ring R and (F , A) a soft set over R with A = [0, 1].

Then (F,A) is a quasi-idealistic soft ring over R if and only if µ is a fuzzy quasi-ideal ofR.

(iv) Let µ be a fuzzy set in a ring R and (Fq, A) a soft set over R with A = [0, 1].Then (Fq, A) is a quasi-idealistic soft ring over R if and only if µ is a fuzzy quasi-idealof R.

Same as in [10], we have the following definitions and lemmas.

4.28. Definition. A fuzzy set µ is said to be an (∈,∈ ∨q)-fuzzy quasi-ideal of R if forall t, r ∈ (0, 1], the following conditions hold for x, y ∈ R:

(P1) xt ∈ µ and yr ∈ µ imply (x− y)min(t,r) ∈ ∨qµ,(P2) xt ∈ (µ ∗ χR)

⋂(χR ∗ µ)implies xt ∈ ∨qµ.

The following lemma follows from the definition.

4.29. Lemma. A fuzzy set µ in a ring R is an (∈,∈ ∨q)-fuzzy quasi-ideal of R if andonly if the following conditions hold for x, y ∈ R:

(Q1) µ(x− y) ≥ min{µ(x), µ(y), 0.5},(Q2) µ(x) ≥ min{((µ ∗ χR)

⋂(χR ∗ µ))(x), 0.5)}.

4.30. Definition. A fuzzy set µ is said to be an (∈,∈ ∨ q)-fuzzy quasi-ideal of R if forall t, r ∈ (0, 1] and x, y ∈ R, the following conditions hold:

(R1) (x− y)min(t,r)∈µ, implies xt∈ ∨ qµ or yr∈ ∨ qµ,(R2) xt∈µ, implies xt∈ ∨ q(µ ∗ χR)

⋂(χR ∗ µ).

The proof of the following lemma is easy and is hence omitted.

4.31. Lemma. A fuzzy set µ in a ring R is an (∈,∈ ∨ q)-fuzzy quasi-ideal of R if andonly if for all x, y ∈ R, the following conditions hold:

(S1) max{µ(x− y), 0.5} ≥ min{µ(x), µ(y)},(S2) max{µ(x), 0.5} ≥ ((µ ∗ χR)

⋂(χR ∗ µ))(x).


4.32. Definition. Let α, β ∈ (0, 1] and α < β. Then a fuzzy set µ is called an (α, β)-fuzzy quasi-ideal of R if for x, y ∈ R, the following conditions hold:

(T1) max{µ(x− y), α} ≥ min{µ(x), µ(y), β},(T2) max{µ(x), α} ≥ min{((µ ∗ χR)

⋂(χR ∗ µ))(x), β}.

The proofs of the following theorem follow from Theorem 3.9, Theorem 3.12, Theorem3.13, Theorem 3.14 and Theorem 3.16, respectively.

4.33. Theorem. (i) Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (F , A) isa quasi-idealistic soft ring over R if and only if µ is an (∈,∈ ∨q)-fuzzy quasi-ideal of R.

(ii) Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (F , A) is a quasi-idealisticsoft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy quasi-ideal of R.

(iii) Let µ be a fuzzy set in a ring R and A = (0, 0.5]. Then (Fq, A) is a quasi-idealisticsoft ring over R if and only if µ is an (∈,∈ ∨ q)-fuzzy quasi-ideal of R.

(iv) Let µ be a fuzzy set in a ring R and A = (0.5, 1]. Then (Fq, A) is a quasi-idealisticsoft ring over R if and only if µ is an (∈,∈ ∨q)-fuzzy quasi-deal of R.

(v) Let µ be a fuzzy set in a ring R and A = (α, β]. Then (F , A) is an quasi-idealisticsoft ring over R if and only if µ is an (α, β)-fuzzy quasi-ideal of R.

5. Soft regular rings

5.1. Definition ([7]). A ring R is called regular if for each element a of R, there existsan element x such that a = axa.

5.2. Definition. A soft ring (F,A) over R is called regular if for ∀x ∈ A, F (x) is regular.

5.3. Example. In example 4.2, (F,A) is a regular soft ring.

5.4. Definition. A ring R is called soft regular if every soft ring (F,A) over R is aregular soft ring.

5.5. Example. Let R = Z6 = {0, 1, 2, 3, 4, 5}. Then every subring of R is regular, andso every soft ring (F,A) over R is a regular soft ring. Thus, R is soft regular.

We now characterize the regular rings by using soft sets.

5.6. Theorem. A ring R is regular if and only if (F,A) ∗ (G,B) = (F,A) ∧ (G,B) forevery right idealistic soft ring (F,A) over R and every left idealistic soft ring (G,B) overR.

Proof. Assume that R is a regular ring. Let (F,A) and (G,B) be any right idealistic softring over R and any left idealistic soft ring over R, respectively. Then for all x ∈ A andfor all y ∈ B, F (x) is a right ideal of R and G(y) is a left ideal of R. Let a ∈ F (x)∩G(y).Then there exists r ∈ R such that a = ara ∈ F (x)G(y). Thus F (x) ∩G(y) ⊂ F (x)G(y).On the other hand, if a ∈ F (x)G(y), then a = bc, b ∈ F (x), c ∈ G(y), and hence,there exist r, s ∈ R such that a = brbcsc. In this case, we have a ∈ F (x) ∩ G(y) andF (x)G(y) ⊂ F (x) ∩ G(y). It hence follows that F (x)G(y) = F (x) ∩ G(y) for every(x, y) ∈ A×B, that is, (F,A) ∗ (G,B) = (F,A) ∧ (G,B).

Conversely, if we let a be an element of R such that F (x) = aR for all x ∈ A andG(y) = Ra for all y ∈ B, then (F,A) is a right idealistic soft ring over R and (G,B) is aleft idealistic soft ring over R. Since (F,A) ∗ (G,B) = (F,A) ∧ (G,B), aR ∩Ra = aRRaand hence a ∈ aR ∩Ra. Thus a ∈ aRRa ⊂ aRa. This shows that R is regular. �

5.7. Lemma ([15]). A ring R is regular if and only if Q = QRQ for every quasi-idealQ of R.


5.8. Theorem. If R is a ring and (F,A)∧ (G,B)∧ (F,A) = (F,A) ∗ (G,B) ∗ (F,A) forevery quasi-idealistic soft ring (F,A) over R, where (G,B) is an absolute soft ring overR, then R is regular.

Proof. Let Q be any quasi-ideal of R and F (x) = Q for all x ∈ A. Then (F,A) is aquasi-idealistic soft ring over R. Since (F,A)∧ (G,B)∧ (F,A) = (F,A) ∗ (G,B) ∗ (F,A),Q = QRQ. It follows that R is regular. �

5.9. Corollary. If R is a ring and (F,A)∧ (G,B)∧ (F,A) = (F,A) ∗ (G,B) ∗ (F,A) forevery bi-idealistic soft ring (F,A) over R, where (G,B) is an absolute soft ring over R,then R is regular.

5.10. Theorem. If R is a regular ring and (G,B) is an absolute soft ring over R, then(F,A)∧(G,B)∧(H,C) = (F,A)∗(G,B)∗(H,C) for every right idealistic soft ring (F,A)over R and every left idealistic soft ring (H,C) over R.

Proof. For all y ∈ B with G(y) = R, let (F,A) and (H,C) be any right idealistic soft ringover R and left idealistic soft ring over R, respectively. Then, for all x ∈ A and z ∈ C,we have F (x)RH(z) ⊂ F (x)RR ⊂ F (x) and F (x)RH(z) ⊂ RRH(z) ⊂ H(z). Hence, wededuce that F (x)RH(z) ⊂ F (x) ∩H(z) and (F,A) ∗ (G,B) ∗ (H,C)⊂(F,A) ∧ (G,B) ∧(H,C).

On the other hand, let x ∈ A and z ∈ C. Since R is regular, ∀a ∈ F (x) ∩ R ∩H(z) ⊂ R, there exists r ∈ R such that a = ara ∈ F (x)RH(z). Hence, we deduce thatF (x) ∩ R ∩ H(z) ⊂ F (x)RH(z) and (F,A) ∧ (G,B) ∧ (H,C)⊂(F,A) ∗ (G,B) ∗ (H,C).Thus, (F,A) ∧ (G,B) ∧ (H,C) = (F,A) ∗ (G,B) ∗ (H,C). �

5.11. Lemma ([15]). A ring R is regular if and only if I ∩ Q = QIQ holds for everyideal I of R and every quasi-ideal Q of R.

5.12. Theorem. If (F,A) ∧ (G,B) ∧ (F,A) = (F,A) ∗ (G,B) ∗ (F,A) holds for everyquasi-idealistic soft ring (F,A) over R and every idealistic soft ring (G,B) over R, thenthe ring R is regular.

Proof. Assume that I is an ideal ofR andQ is a quasi-ideal ofR. If F (x) = Q for all x ∈ Aand G(y) = I for all y ∈ B, then (F,A) is a quasi-idealistic soft ring over R and (G,B)is an idealistic soft ring over R. Since (F,A) ∧ (G,B) ∧ (F,A) = (F,A) ∗ (G,B) ∗ (F,A),I ∩Q = QIQ. This shows that R is regular. �

5.13. Corollary. If (F,A) ∧ (G,B) ∧ (F,A) = (F,A) ∗ (G,B) ∗ (F,A) holds for everybi-idealistic soft ring (F,A) and every idealistic soft ring (G,B) over a ring R, then R isa regular ring.

5.14. Theorem. If R is a regular ring, then (F,A)∧ (G,B)∧ (H,C) = (F,A) ∗ (G,B) ∗(H,C) holds for every right idealistic soft ring (F,A) over R, every idealistic soft ring(G,B) over R and every left idealistic soft ring (H,C) over R.

Proof. Let (F,A), (G,B) and (H,C) be a right idealistic soft ring over R, an idealistic softring over R and a left idealistic soft ring over R, respectively. Then for all x ∈ A, for ally ∈ B, and for all z ∈ C, we have F (x)G(y)H(z) ⊂ F (x)RR ⊂ F (x), F (x)G(y)H(z) ⊂RRH(z) ⊂ H(z) and F (x)G(y)H(z) ⊂ RG(y)R ⊂ G(y). Hence, F (x)G(y)H(z) ⊂F (x) ∩G(y) ∩H(z).

On the other hand, if for all a ∈ F (x)∩G(y)∩H(z) ⊂ R, there exists r ∈ R such thata = ara = arara ∈ F (x)G(y)H(z), then this leads to F (x)G(y)H(z) ⊃ F (x) ∩ G(y) ∩H(z). Hence, F (x)G(y)H(z) = F (x) ∩ G(y) ∩ H(z) and so (F,A) ∧ (G,B) ∧ (H,C) =(F,A) ∗ (G,B) ∗ (H,C). �


5.15. Theorem. For a ring R, the following conditions are equivalent:(1) R is regular.(2) (F,A) ∧ (G,B)⊂(F,A) ∗ (G,B) for every right idealistic soft ring (F,A) over R

and every bi-idealistic soft ring (G,B) over R.(3) (F,A) ∧ (G,B)⊂(F,A) ∗ (G,B) for every right idealistic soft ring (F,A) over R

and every quasi-idealistic soft ring (G,B) over R.(4) (F,A) ∧ (H,C)⊂(F,A) ∗ (H,C) for every bi-idealistic soft ring (F,A) over R and

every left idealistic soft ring (H,C) over R.(5) (F,A) ∧ (H,C)⊂(F,A) ∗ (H,C) for every quasi-idealistic soft ring (F,A) over R

and every left idealistic soft ring (H,C) over R.(6) (F,A)∧ (G,B)∧ (H,C)⊂(F,A)∗ (G,B)∗ (H,C) for every right idealistic soft ring

(F,A) over R, every bi-idealistic soft ring (G,B) over R and every left idealistic soft ring(H,C) over R.

(7) (F,A)∧ (G,B)∧ (H,C)⊂(F,A)∗ (G,B)∗ (H,C) for every right idealistic soft ring(F,A) over R, every quasi-idealistic soft ring (G,B) over R and every left idealistic softring (H,C) over R.

Proof. Assume that (1) holds. Let x ∈ A, y ∈ B. Since R is regular, for all a ∈F (x) ∩G(y) ⊂ R, there exists r ∈ R such that a = ara = (ar)a ∈ F (x)G(y). This leadsto (F,A) ∧ (G,B)⊂(F,A) ∗ (G,B) and so (2) holds. Thus,(1) implies (2).

It can be similarly proved that (1) implies (4). Since any quasi-idealistic soft ring overR is a bi-idealistic soft ring over R, (2) also implies (3), and (4) implies (5).

Assume that (3) holds. Since any left idealistic soft ring over R is a quasi-idealisticsoft ring, by Theorem 5.6, R is regular, and so (3) implies (1).

Similarly, we can prove (5) implies (1).Assume that (1) holds. Let (F,A), (G,B) and (H,C) be any right idealistic soft ring

over R, any bi-idealistic soft ring over R and any left idealistic soft ring over R. Letx ∈ A, y ∈ B and z ∈ C. Since R is regular, for all a ∈ F (x) ∩ G(y) ∩ H(z) ⊂ R,there exists r ∈ R such that a = ara = arara = (ar)a(ra) ∈ F (x)G(y)G(z). Hence,(F,A) ∧ (G,B) ∧ (H,C)⊂(F,A) ∗ (G,B) ∗ (H,C) and (6) holds. Thus (1) implies (6).

It is clear that (6) implies (7).Finally, we assume that (7) holds. Let (F,A) and (H,C) be any right idealistic soft

ring over R and any left idealistic soft ring over R, respectively. If (G,B) is an absolutesoft ring over R, then (G,B) is a quasi-idealistic soft ring over R. This implies that(F,A)∧ (G,B)∧ (H,C)⊂(F,A)∗ (G,B)∗ (H,C). Let x ∈ A, z ∈ C. Then F (x)∩H(z) =F (x)∩R∩H(z) ⊂ F (x)RH(z) ⊂ F (x)H(z) and (F,A)∧ (H,C)⊂(F,A) ∗ (H,C). Hence,it follows that R is regular and so (7) implies (1). �

Finally, we state the following theorem of regular rings to be soft regular.

5.16. Theorem. If a ring R is soft regular, then R is regular.

Proof. If (F,A) is an absolute soft ring over R, then F (x) = R is a regular ring. �

5.17. Corollary. If a ring R is soft regular, then (F,A) ∗ (G,B) = (F,A) ∧ (G,B) forevery right idealistic soft ring (F,A) over R and every left idealistic soft ring (G,B) overR.

5.18. Corollary. If R is a soft regular ring and (G,B) is an absolute soft ring over R,then (F,A)∧ (G,B)∧ (H,C) = (F,A) ∗ (G,B) ∗ (H,C) for every right idealistic soft ring(F,A) over R and every left idealistic soft ring (H,C) over R.

5.19. Corollary. If R is a soft regular ring, then (F,A) ∧ (G,B) ∧ (H,C) = (F,A) ∗(G,B) ∗ (H,C) for every right idealistic soft ring (F,A) over R, every idealistic soft ring(G,B) over R and every left idealistic soft ring (H,C) over R.


5.20. Corollary. If a ring R is soft regular, then the following conditions hold:(1) (F,A) ∧ (G,B)⊂(F,A) ∗ (G,B) for every right idealistic soft ring (F,A) over R

and every bi-idealistic soft ring (G,B) over R.(2) (F,A) ∧ (G,B)⊂(F,A) ∗ (G,B) for every right idealistic soft ring (F,A) over R

and every quasi-idealistic soft ring (G,B) over R.(3) (F,A) ∧ (H,C)⊂(F,A) ∗ (H,C) for every bi-idealistic soft ring (F,A) over R and

every left idealistic soft ring (H,C) over R.(4) (F,A) ∧ (H,C)⊂(F,A) ∗ (H,C) for every quasi-idealistic soft ring (F,A) over R

and every left idealistic soft ring (H,C) over R.(5) (F,A)∧ (G,B)∧ (H,C)⊂(F,A) ∗ (G,B) ∗ (H,C) for for every right idealistic soft

ring (F,A) over R, every bi-idealistic soft ring (G,B) over R and every left idealistic softring (H,C) over R.

(6) (F,A)∧ (G,B)∧ (H,C)⊂(F,A)∗ (G,B)∗ (H,C) for every right idealistic soft ring(F,A) over R, every quasi-idealistic soft ring (G,B) over R and every left idealistic softring (H,C) over R.

In order to answer when will a regular ring be soft regular, we give the followinglemma.

5.21. Lemma. If a ring R is regular, then every idealistic soft ring over R is regular.

Proof. Let (F,A) be any idealistic soft ring over R. Then ∀x ∈ A, F (x) is an ideal of R.If a ∈ F (x), then a ∈ R and there exists an element r ∈ R such that a = ara = arara =a(rar)a ∈ aF (x)a. Thus F (x) is regular and (F,A) is a regular soft ring over R. �

By using the above lemma, we obtain the following theorem for regular rings to besoft regular.

5.22. Theorem. If a ring R is regular and every soft ring (F,A) over R is an idealisticsoft ring, then R is soft regular.

Proof. If (F,A) is a soft ring over R, then (F,A) is an idealistic soft ring over R. Hence,by Lemma 5.21, (F,A) is regular and consequently R is soft regular. �

Acknowledgements

The authors are highly grateful to the anonymous reviewers for their helpful commentsand suggestions for improving the paper.

This research is partially supported by a grant of National Natural Science Foundationof China (61175055), Innovation Term of Higher Education of Hubei Province, China(T201109), Natural Science Foundation of Hubei Province (2012FFB01101), NaturalScience Foundation of Education Committee of Hubei Province, China (B20122904) andInnovation Term of Hubei University for Nationalities (MY2012T002).

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quium, 19, 391-397, 2012.

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ABSOLUTE CO-SUPPLEMENT AND

ABSOLUTE CO-COCLOSED MODULES

Derya Keskin Tutuncu∗ and Sultan Eylem Toksoy†


Abstract

A module M is called an absolute co-coclosed (absolute co-supplement)module if whenever M ∼= T/X the submodule X of T is a co-closed (supplement) submodule of T . Rings for which all modulesare absolute co-coclosed (absolute co-supplement) are precisely deter-mined. We also investigate the rings whose (finitely generated) abso-lute co-supplement modules are projective. We show that a commu-tative domain R is a Dedekind domain if and only if every submod-ule of an absolute co-supplement R-module is absolute co-supplement.We also prove that the class Coclosed of all short exact sequences0 //A //B //C //0 such that A is a coclosed submoduleof B is a proper class and every extension of an absolute co-coclosedmodule by an absolute co-coclosed module is absolute co-coclosed.

Keywords: Absolute co-supplement (co-coclosed) module, Supplement (coclosed) sub-module.

2000 AMS Classification: Primary 16 D 10. Secondary 06 C 05.

1. Preliminaries

Throughout this paper unless otherwise stated all rings are associative with identityelement and all modules are unitary right R-modules. A submodule N of M is said tobe coclosed if N/K � M/K implies K = N for each K ≤ N or equivalently, given anyproper submodule K of N , there is a submodule L of M for which N + L = M butK + L 6= M . A submodule K of M is said to be supplement of N in M if K is minimalwith respect to K+N = M equivalently, K+N = M and K ∩N � K. A submodule Lof M is called a supplement submodule in M provided there exists a submodule X of Msuch that L is a supplement of X in M . A module M is said to be supplemented, if everysubmodule of M has a supplement in M . Every supplement submodule of a module M is

∗Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey. E-mail:

[email protected]†Department of Mathematics, Izmir Institute of Technology, 35430 Urla, Izmir, Turkey. E-

mail: [email protected]

68 D.Keskin Tutuncu, S. E. Toksoy

coclosed (see, for example [3, 20.2]). A module M is called absolute supplement (or almostinjective) if it is a supplement submodule of every module containing M (see [2] and [5]).If for all short exact sequences 0 //X //T //M //0 the submodule X isa supplement submodule of T , then M is said to be absolute co-supplement (see [5]).Clearly if M/N is absolute co-supplement, then N is a supplement submodule of M .

In this paper we introduce the notion of absolute co-coclosed modules. If for all shortexact sequences 0 //X //T //M //0 the submodule X is coclosed in T ,then M is called an absolute co-coclosed module. Clearly if M/N is absolute co-coclosed,then N is a coclosed submodule of M . Since supplement submodules are coclosed, thefollowing implication hold:

absolute co-supplement ⇒ absolute co-coclosed

In section 2 we give a characterization and some properties of absolute co-supplementmodules. We also characterize the rings whose modules are absolute co-supplement.For example, R is semisimple if and only if every R-module is absolute co-supplement(see Theorem 2.8). We also investigate the rings whose (finitely generated) absoluteco-supplement modules are projective (see Theorem 2.14). We prove that if R is aright hereditary ring, then every absolute co-supplement right R-module is projective(Theorem 2.15). We show that a module M is projective if and only if M is absoluteco-supplement and flat, over any ring R (Theorem 2.18). Let

0 //X //P //M //0

be an exact sequence of finitely generated right R-modules with P projective and J theJacobson radical of R. Then M is absolute co-supplement if and only if the inducedsequence 0 //X/XJ //P/PJ //M/MJ //0 is split exact if and only if

whenever Y is a right R-module with Y J = 0, Ext1R(M,Y ) = 0 (Theorem 2.22). Fi-

nally, we prove that a commutative domain R is a Dedekind domain if and only if everysubmodule of an absolute co-supplement R-module is absolute co-supplement (Theorem2.26).

In section 3 we prove that every right R-module is absolute co-coclosed if and onlyif R is a right V -ring (Theorem 3.6). We show that the class of absolute co-coclosedmodules contains properly the class of absolute co-supplement modules (Example 3.7).We also show that the class of short exact sequences 0 //A //B //C //0with the property that A is coclosed in B is a proper class (Theorem 3.11) and so weinvestigate some properties of absolute co-coclosed modules.

[14] is the general reference for notions of modules not defined in this work.

2. Absolute Co-supplement Modules

Let P be a class of short exact sequences of R-modules and R-module homomorphisms.If a short exact sequence

E : 0 //Af //B

g //C //0

belongs to P, then f is said to be P-monomorphism and g is said to be P-epimorphism.The class P is said to be proper if it satisfies the following conditions (see [15, Introduc-tion]):

(P1) If a short exact sequence E is in P, then P contains every short exact sequenceisomorphic to E .

(P2) P contains all splitting short exact sequences.(P3) The composite of two P-monomorphisms is a P-monomorphism if this composite

is defined.

Absolute Co-Supplement and Absolute Co-Coclosed Modules 69

(P3′) The composite of two P-epimorphisms is a P-epimorphism if this composite isdefined.

(P4) If g and f are monomorphisms, and g ◦ f is a P-monomorphism, then f is aP-monomorphism.

(P4′) If g and f are epimorphisms, and g ◦ f is a P-epimorphism, then g is a P-epimorphism.

2.1. Theorem. (see [7, Theorem 1] or [5, Theorem 3.1.2]) The class

Suppl = {E : 0 //Af //B

g //C //0 |A is a supplement inB}

is a proper class.

A class M of modules is said to be closed under extensions if U,M/U ∈ M impliesM ∈ M. In this case M is an extension of U by M/U . The following proposition showsthat the class of absolute co-supplement modules are closed under extensions.

2.2. Proposition. (see [5, Proposition 3.2.7]) For a module M , if a submodule N andthe quotient module M/N of M are absolute co-supplement, then M is also absoluteco-supplement.

2.3. Theorem. (see [5, Proposition 3.2.2]) For a module M the following conditions areequivalent:

(i) M is an absolute co-supplement module, i.e. for all short exact sequences

0 //X //T //M //0

X is a supplement submodule of T .(ii) There exists a short exact sequence

0 //N //P //M //0

with a projective (or free) module P such that N is a supplement submodule ofP .

Absolute co-supplement and absolute supplement do not imply each other. Clearlyevery projective module is absolute co-supplement. Therefore Z is an absolute co-supplement Z-module. Note that since Z is not a supplement in Q, Z is not absolutesupplement.

An arbitrary factor module of an absolute co-supplement (absolute co-coclosed) mo-dule need not be absolute co-supplement (absolute co-coclosed) (for example, Z/nZ withn nonzero), but we have the following:

2.4. Proposition. (see [5, Proposition 3.2.6]) If M is an absolute co-supplement moduleand N is a supplement submodule of M then M/N is also absolute co-supplement.

2.5. Corollary. Let M1,M2, . . . ,Mn be modules. Then M = M1 ⊕M2 ⊕ . . . ⊕Mn isabsolute co-supplement if and only if each Mi is absolute co-supplement.

Proof. (⇒) : By Proposition 2.4, Mi∼= M/

n⊕j 6=i

Mj is absolute co-supplement for each i.

(⇐) : By Proposition 2.2 and induction. �

2.6. Lemma. Let M be any module. If for every small submodule N of M the factormodule M/N is absolute co-coclosed, then Rad(M) = 0.

Proof. Let N be a small submodule of M . Since M/N is absolute co-coclosed, N is acoclosed submodule of M . Thus N = 0. Therefore Rad(M) = 0. �


2.7. Proposition. Let M be a module. If every factor module of M is absolute co-supplement, then M is semisimple.

Proof. In this case every submodule of M is a supplement. By Lemma 2.6, Rad(M) = 0.Hence M is semisimple. �

2.8. Theorem. The following are equivalent for a ring R:

(i) Every right R-module is absolute co-supplement.(ii) Every factor module (submodule) of every right R-module is absolute co-supplement.

(iii) Every factor module of every projective right R-module is absolute co-supplement.(iv) Every factor module of every free right R-module is absolute co–supplement.(v) Every factor module of RR is absolute co-supplement.

(vi) R is semisimple.(vii) Every right R-module is absolute supplement.

(viii) The analogues of the above properties for left R-modules.

Proof. (i)⇔ (ii) and (ii)⇒ (iii)⇒ (iv)⇒ (v) are clear.(v)⇒ (vi) : By Proposition 2.7.(vii)⇔ (vi) : By [2, Proposition 2.9].(vi)⇒ (i) : Clear.(vi)⇔ (viii) : By the left right symmetry of semisimple rings. �

2.9. Theorem. (see [11, Theorem 3.7.2]) Let R be a ring with J = 0, where J is theJacobson radical of R. An R-module M is projective if and only if M is absolute co-supplement.

As a consequence:

2.10. Corollary. Let R be a regular (or a right V -)ring. Then any module M is projec-tive if and only if it is absolute co-supplement.

2.11. Lemma. Let R be any ring. Then the following are equivalent:

(i) In every projective right R-module, every supplement is a direct summand.(ii) Every absolute co-supplement right R-module is projective.

Proof. (i) ⇒ (ii) : Let M be an absolute co-supplement right R-module. Then thereexists a projective module P such that P/X ∼= M for some supplement submodule X ofP . By (1), X is a direct summand of P . Thus M is projective.(ii) ⇒ (i) : Let P be a projective right R-module and X be a supplement submoduleof P . By (ii) and Proposition 2.4, P/X is projective and so X is a direct summand ofP . �

2.12. Lemma. Let R be any ring. Then the following are equivalent:

(i) In every finitely generated projective right R-module, every supplement submod-ule is a direct summand.

(ii) Every finitely generated absolute co-supplement right R-module is projective.(iii) The analogues of the above properties for left R-modules.

Proof. (i)⇔ (ii) : By the same proof as (i)⇔ (ii) of Lemma 2.11.(i)⇔ (iii) : By [14, Proposition A9] or [18, Corollary of Theorem 2.3]. �

2.13. Remark. Let R be any ring and J its Jacobson radical. Then the followingstatements are equivalent (see [13], [14] and [18]):

(1) If M is a finitely generated flat right R-module and M/MJ is a projective R/J-module, then M is a projective right R-module.


(2) If G is a projective right R-module and G/GJ is finitely generated, then G isfinitely generated.

(3) If Q is a projective right R-module, then each finitely generated proper submod-ule is contained in a maximal submodule.

(4) Every supplement in a finitely generated projective right R-module is a directsummand.

(5) Every finitely generated J-projective right R-module (a module P is J-projectiveprovided wheneverX −→ Y is an epimorphism with Y J = 0, then Hom(P,X) −→Hom(P, Y ) is an epimorphism) is projective.

(6) Every finitely presented J-projective right R-module is projective.(7) If M is a finitely presented right R-module such that M/MJ is projective and

Tor1R(R/J,M) = 0, then M is projective.

(8) If M is a finitely presented right R-module such that Ext1R(M,Y ) = 0 for all

modules Y with Y J = 0, then M is projective.(9) The analogues of the above properties for left R-modules.

All these properties are satisfied for commutative rings, as well as for rings such thatevery prime factor ring is right (or left) Goldie (in particular, for right or left noetherianrings, and for rings with a polynomial identity).

If R is a commutative domain or a right noetherian ring, then (4) is valid for every(not necessarily finitely generated) projective R-module.

Note that Zoschinger called any ring satisfying the condition (2) right-L-ring andshowed that this notion is left-right symmetric (see [18]).

In the light of Remark 2.13 we have the following theorem:

2.14. Theorem. (1) Let M be an absolute co-supplement module. Then M is pro-jective in each of the following cases for R:(a) R is a commutative domain.(b) R is a right noetherian ring.

(2) Let M be a finitely generated absolute co-supplement module. Then M is pro-jective in each of the following cases for R:(a) R is commutative.(b) R is a ring such that every prime factor ring is right (or left) Goldie.(c) R is a right or left noetherian ring.(d) R is a ring with polynomial identity.

Proof. By Remark 2.13 and Lemmas 2.11 and 2.12. �

2.15. Theorem. If R is a right hereditary ring, then every absolute co-supplement rightR-module is projective.

Proof. Let M be an absolute co-supplement module. Then there exists a projectivemodule P such that P/X ∼= M for some supplement submodule X of P . By [18, Corollary(1) of Lemma 2.1], M is projective. �

There exists an absolute co-supplement module which is not projective:

2.16. Example. Let R be a ring with the Jacobson radical J . Zoschinger proved in [18,Theorem 1.2] that in RR every supplement is a direct summand if and only if ab = 0 and

1 − (a + b) ∈ J imply b · 1

a+ b· a = 0. (This means that for any commutative ring R

supplements and direct summands in R coincide). What Zoschinger observes is that


(*) if v is in R with vR = v2R and v − v2 ∈ J , then vR is a supplement of (1 − v)R:Since vR+ (1− v)R = R it follows that (v− v2)R = vR∩ (1− v)R. (v− v2) = (1− v)v =(1− v)v2t = v(v − v2)t ∈ vJ . Since J � R, (v − v2)R is small in vR.

Note that if t is inR with v2t = v, then (v−v2)t = vt−v ∈ J . Moreover v(1−(vt−v)) =v2. So that u = 1− (vt− v) satisfies that v = v2u−1. Therefore (*) is equivalent to

(**) If v is in R with v − v2 ∈ J such that there exists a unit u ∈ R with u ∈ 1 + Jand v = v2u−1, then vR is a supplement of (1− v)R.

When this happens on the right it also happens on the left.Note that v is idempotent modulo J . Having (**) is equivalent to have a projective

left module that modulo J is isomorphic to the projective R/J-module generated byv + J which in turn is equivalent to have a projective right module that modulo J isisomorphic to the projective R/J-module generated by 1− v + J .

The Gerasimov-Sakhaev example in [8] was the first showing that for a semilocal ring(**) could be satisfied without v being idempotent.

Let k be a field. Take a free algebra on two generators k < x, y >, and let ϕ : k <x, y >→ k× k be defined by ϕ(x) = (1, 0) and ϕ(y) = (0, 1). Since ϕ(y)ϕ(x) = (0, 0) themap can be factorized to the ring S = k < x, y > /(yx). We call the induced map still ϕ.

Let Σ denote the set of all squared matrices such that the image via ϕ is invertible.Then we can invert these matrices to get a new ring R = SΣ and morphisms λ : S → Rand ϕ′ : R → k × k such that ϕ = ϕ′λ. Note that this implies that ϕ′ is onto. By theconstruction, Ker(ϕ′) = J . So that R is a semilocal ring and R/J ∼= k × k.

By abuse of notation (which is better not to do), we call x, y ∈ R to the images ofx+(yx) and y+(yx) via λ. Since ϕ(x+y+(yx)) = (1, 1) which is a 1×1 invertible matrixover k×k we deduce that u = x+y is invertible in S. Note that y(x+y)+(yx) = y2+(yx)so that y = y2u in R. Note also that y−y2+(yx) is in the kernel of ϕ so that ϕ′(y−y2) = 0and so y − y2 ∈ J . Therefore, yR is a supplement of (1 − y)R in R. Note that in [8],Gerasimov and Sakhaev show that yR is not a direct summand of RR. Therefore theR-module R/yR is absolute co-supplement but not projective over the ring R.

2.17. Example. It is well-known that QZ is flat but not projective, and so it is notabsolute co-supplement by Theorem 2.9.

Therefore we can give the following theorem:

2.18. Theorem. Let M be a module. Then M is absolute co-supplement and flat if andonly if M is projective.

Proof. We only need to prove the necessity. Since M is absolute co-supplement, thereexists a projective module P such that P/X ∼= M for some supplement submodule Xof P . Since P/X is flat, X is a pure submodule of P by [1, Exercise 19(11) or Lemma19.18]. Now by [18, Corollary (2) of Lemma 2.1], X is a direct summand of P . Hence Mis projective. �

2.19. Proposition. If M is a finitely generated absolute co-supplement module, then Mis finitely presented.

Proof. Since M is absolute co-supplement and finitely generated, there is a finitely gen-erated projective module P such that M ∼= P/X where X is a supplement submodule ofP . It is well-known that X is finitely generated since P is finitely generated. ThereforeM is finitely presented. �

2.20. Corollary. Let R be a serial ring. Then every finitely generated absolute co-supplement module is a finite direct sum of local modules.


Proof. By [17, Corollary 3.4] and Proposition 2.19, M is a finite direct sum of localmodules. �

Any finitely presented module need not be absolute co-supplement (for example, Z/nZwith n nonzero). Now we have the following corollary:

2.21. Corollary. Let M be a finitely generated flat module. Then the following areequivalent:

(i) M is projective.(ii) M is absolute co-supplement.

(iii) M is finitely presented.

Proof. (i)⇒ (ii) : Clear.(ii)⇒ (i) : By Theorem 2.18.(ii)⇒ (iii) : By Proposition 2.19.(iii)⇒ (i) : By [10, Theorem 4.30] or [9, Corollary 5.3], M is projective.

�

2.22. Theorem. Let 0 //X //P //M //0 be an exact sequence of finitelygenerated right R-modules with P projective (for example if M is finitely presented) andJ the Jacobson radical of R. Then the following are equivalent:

(i) M is absolute co-supplement.(ii) The induced sequence 0 //X/XJ //P/PJ //M/MJ //0 is split

exact.(iii) If Y is a right R-module with Y J = 0, then Ext1

R(M,Y ) = 0.

Proof. By [13, Theroem 2.6]. �

On the other hand, we have:

2.23. Proposition. Let M be an absolute co-supplement module. Then Ext1R(M,K) = 0

for all modules K with Rad(K) = 0.

Proof. Let 0 //K //A //M //0 be a short exact sequence. Since M isabsolute co-supplement, there exists a submodule L of A such that A = K + L andK ∩ L� K. So A = K ⊕ L. Hence Ext1

R(M,K) = 0. �

It is well known that for a nonzero finitely generated Z-module M , Ext1Z(M,Z) =

T (M), where T (M) is the torsion submodule of M . Now Proposition 2.23 gives us a wellknown result for nonzero finitely generated projective Z-modules: Let M be a nonzerofinitely generated projective Z-module. Then M is torsion-free.

Now we close this section with the following elementary observations:

2.24. Lemma. Let R be a ring. If R is right hereditary, then every submodule of anabsolute co-supplement right R-module is absolute co-supplement.

Proof. By Theorem 2.15. �

2.25. Theorem. Let R be a right noetherian ring. Then R is right hereditary if and onlyif every submodule of an absolute co-supplement right R-module is absolute co-supplement.In this case R is left semihereditary.

Proof. By Lemma 2.24, Theorem 2.14(1)(b) and [10, Corollary 7.65]. �

2.26. Theorem. Let R be a commutative domain. Then R is a Dedekind domain if andonly if every submodule of an absolute co-supplement R-module is absolute co-supplement.


Proof. (⇒) : By Theorem 2.25.(⇐) : By Theorem 2.14(1)(a), R is hereditary. Thus R is a Dedekind domain. �

3. Absolute Co-coclosed Modules

3.1. Lemma. Let M be an absolute co-coclosed module such that it has a projectivecover. Then M is projective.

Proof. There exists a projective module P and a submodule K of P such that K is smallin P and M ∼= P/K. Since M is absolute co-coclosed, K is a coclosed submodule of P .Thus K = 0. Hence M is projective. �

3.2. Proposition. Let R be a right perfect (semiperfect) ring. Then the following areequivalent for a (finitely generated) module M :

(i) M is projective.(ii) M is absolute co-supplement.

(iii) M is absolute co-coclosed.

Proof. By Lemma 3.1. �

Recall that any module M is co-semisimple (or a V -module) if every simple R-moduleis M -injective. Equivalently, Rad(M/T ) = 0 for every submodule T of M (see [4, 2.13]).

3.3. Lemma. (see [3, 3.8]) Let M be any module. Then M is a V -module if and only ifevery submodule of M is coclosed.

3.4. Proposition. Let M be a module. If every factor module of M is absolute co-coclosed, then M is a V -module.

Proof. In this case every submodule of M is coclosed. By Lemma 3.3, M is a V -module.�

By [4, 2.13], we know that any ring R is a right V -ring if and only if every rightR-module is a V -module. Thus we have the following lemma:

3.5. Lemma. (see [6, Proposition 2.1]) Let R be a ring. R is a right V -ring if and onlyif for any R-module M , every submodule of M is coclosed in M .

3.6. Theorem. Let R be a ring. The following are equivalent:

(i) Every right R-module is absolute co-coclosed.(ii) Every factor module (submodule) of every right R-module is absolute co-coclosed.

(iii) Every factor module of every projective right R-module is absolute co-coclosed.(iv) Every factor module of every free right R-module is absolute co-coclosed.(v) Every factor module of RR is absolute co-coclosed.

(vi) R is a right V -ring.

Proof. (i)⇔ (ii) and (ii)⇒ (iii)⇒ (iv)⇒ (v) are clear.(v)⇒ (vi) : By Proposition 3.4.(vi) ⇒ (i) : Assume that R is a right V -ring. Let M be any R-module. Let M ∼= T/Xfor a module T and a submodule X of T . By Lemma 3.5, X is coclosed in T . Thus Mis absolute co-coclosed. �

3.7. Example. There exists an absolute co-coclosed module M which is not absoluteco-supplement over a right V -ring which is not semisimple by Theorems 3.6 and 2.8. Forexample, let K be a field and let R =

∏n≥1

Kn with Kn = K for all n ≥ 1. Then the

ring R is a commutative von Neumann regular ring (namely it is a V -ring) which is not


semisimple. Note that A =⊕n≥1

Kn is not a direct summand of R. By Theorem 3.6, R/A

is an absolute co-coclosed R-module. By Corollary 2.10 it is not absolute co-supplement.

As an easy observation we can give the following lemma:

3.8. Lemma. Let M be an absolute co-coclosed module. Then every epimorphism fromA to M with small kernel is an isomorphism for any module A.

3.9. Proposition. Let M be a quasi-discrete module. If M is absolute co-coclosed, thenM is discrete.

Proof. By Lemma 3.8 and [14, Lemma 5.1]. �

Note that any discrete module need not be absolute co-coclosed (Z/pZ, where p is anyprime).

We announce that the following lemma was proved by Zoschinger in [19]. In this notewe give his proof for the completeness.

3.10. Lemma. (see [19, Lemma A4]) Let U ≤ V ≤ M be submodules of M . If U iscoclosed in M and V/U is coclosed in M/U , then V is coclosed in M .

Proof. Let X ≤ V and V/X �M/X. Firstly, we will prove that U/(X ∩U)�M/(X ∩U). Let X ∩ U ≤W ≤M with W + U = M :Step 1: V/[U + (W ∩ X)] � M/[U + (W ∩ X)]: Let U + (W ∩ X) ≤ Z ≤ M andZ + V = M . Then (Z ∩W ) + V = M . Since V/X is small in M/X, (Z ∩W ) +X = Mand since (Z ∩W ) + (X ∩W ) = W , W ≤ Z. Finally since Z +W = M , Z = M .Step 2: Since V/U is coclosed in M/U , U + (W ∩X) = V . Now (U ∩X) + (W ∩X) = X,and X ≤W . Since V/X �M/X, W = M . �

By using Lemma 3.10 the following Theorem 3.11 can be obtained easily.

3.11. Theorem. The class

Coclosed = {E : 0 //Af //B

g //C //0 |A is coclosed inB}is a proper class.

A module C is said to be coprojective relative to a proper class P (or P-coprojective) ifevery epimorphism B −→ C is a P-epimorphism. With the help of (P3) it can be shownthat if 0 //A //B //C //0 is a short exact sequence in P with the modulesA and C both P-coprojective, then so is B [12, Proposition 1.9]. A module C is a P-coprojective module if and only if there is a P-epimorphism of a projective module B ontoC [12, Proposition 1.12]. When (P3′) is taken into account it follows that the image of aP-coprojective module under a P-epimorphism is always P-coprojective [12, Proposition1.13]. In this paper Coclosed-coprojective (Suppl-coprojective) modules have been calledabsolute co-coclosed (absolute co-supplement) modules. As a result of above informationProposition 3.12, Theorem 3.13 and Corollary 3.14 are obtained immediately. Here wegive their proofs for completeness. For similar results on absolute co-supplement modulessee [5].

3.12. Proposition. Every extension of an absolute co-coclosed module by an absoluteco-coclosed module is absolute co-coclosed.

Proof. Let N ≤ M such that N and M/N are absolute co-coclosed. We want to showthat M is absolute co-coclosed. Let 0 //X //T //M //0 be any short


exact sequence. We have the following diagram:

0

��

0

��X

f

��

= X

g◦f

��0 // K

g//

��

T

��

// M/N //

‖

0

0 // N //

��

M //

��

M/N // 0

0 0

Since N is absolute co-coclosed, f is a Coclosed-monomorphism and since M/N isabsolute co-coclosed, g is a Coclosed-monomorphism. Therefore g ◦ f is a Coclosed-monomorphism by Theorem 3.11. Thus M is an absolute co-coclosed module. �

3.13. Theorem. For a module M the following conditions are equivalent:

(i) M is an absolute co-colcosed module.(ii) There exists a projective (or free) module P with M ∼= P/N such that N is

coclosed in P .

Proof. (i) ⇒ (ii) : For every module M , there is a projective (or free) module P andan epimorphism f from P to M . So M ∼= P/Ker f and by the definition of an absoluteco-coclosed module Ker f is coclosed in P .

(ii) ⇒ (i) : Let 0 //X //T α //M //0 be a short exact sequence. By (ii),

there is a short exact sequence 0 //N //Pg //M //0 with a projective mo-

dule P such that N is coclosed in P . So we have the following diagram:

0

��

0

��N =

��

N

��0 // X //

‖

Yf //

β

��

P //

g

��

0

0 // X // T α //

��

M //

��

0

0 0

where Y together with homomorphisms f and β is a pullback of the pair of homomor-phisms g and α. Since P is projective, f is a splitting epimorphism. So α ◦ β = g ◦ f is aCoclosed-epimorphism. Then α is a Coclosed-epimorphism by Theorem 3.11. ThereforeM is an absolute co-coclosed module. �


3.14. Corollary. If N is a coclosed submodule of an absolute co-coclosed module M ,then M/N is also absolute co-coclosed.

Proof. Since N is coclosed in M , the short exact sequence

0 //N //M σ //M/N //0

is in the class Coclosed. Since M is an absolute co-coclosed module, there exists a shortexact sequence

0 //K //Pf //M //0

with a projective module P such that K is coclosed in P by Theorem 3.13. Then wehave the following diagram:

0

��

0

��K

��

= K

��0 // T //

��

P

f

��

σ◦f // M/N //

‖

0

0 // N //

��

Mσ //

��

M/N // 0

0 0

where T = Ker(σ ◦ f). Now f and σ are Coclosed-epimorphisms, therefore σ ◦ f isalso a Coclosed-epimorphism by Theorem 3.11. So by Theorem 3.13, M/N is absoluteco-coclosed. �

Let M be any module. Talebi and Vanaja define

Z(M) =⋂{Ker g | g ∈ Hom(M,L), L is a small module}.

They call M a non-cosingular module if Z(M) = M (see [16]).

3.15. Corollary. Let M be an absolute co-coclosed module and N ≤M .

(1) If N is a non-cosingular submodule of an absolute co-coclosed module M , thenM/N is also absolute co-coclosed.

(2) If M is, in addition, non-cosingular then the following are equivalent:(i) N is non-cosingular.

(ii) M/N is absolute co-coclosed.(iii) N is coclosed in M .

Proof. (1) Clear by Corollary 3.14 and [16, Lemma 2.3 (2)].(2) (i)⇒ (ii) : By (1).

(ii)⇒ (iii) : By definition.(iii)⇒ (i) : By [16, Lemma 2.3 (3))].

�

Finally we prove the following:


3.16. Proposition. Let M = M1 +M2 such that every factor modules of M1 and M2 areabsolute co-supplement (absolute co-coclosed). Then every factor module of M is absoluteco-supplement (absolute co-coclosed).

Proof. Let N be a submodule of M . Then (M1 + N)/N ∼= M1/(N ∩M1) is absoluteco-supplement (absolute co-coclosed). Since

M/N

(M1 +N)/N∼=

M2

M2 ∩ (M1 +N)

is absolute co-supplement (absolute co-coclosed), by Proposition 2.2 (Proposition 3.12),M/N is absolute co-supplement (absolute co-coclosed). �

Acknowledgements

This paper was written while the second author was visiting Hacettepe Universityas a postdoctoral researcher. She wishes to thank the members of the Department ofMathematics for their kind hospitality and the Scientific and Technical Research Councilof Turkey (TUBITAK) for their financial support. The authors would like to thankProfessor Rafail Alizade who is the PhD advisor of the second author and who hasintroduced the absolute co-supplement concept. The authors also would like to thankProfessor Dolors Herbera and the referee for the valuable comments on the paper.

References

[1] Anderson, F. W. and Fuller, K. R. Rings and Categories of Modules (New York: Springer,

1974.)[2] Clark, J., Keskin Tutuncu D. and Tribak, R. Supplement submodules of injective modules,

Comm. Algebra. 39, 4390–4402, 2011.[3] Clark, J., Lomp, C., Vanaja N. and Wisbauer, R. Lifting Modules, Supplements and Pro-

jectivity in Module Theory (Frontiers in Math. Boston: Birkhauser, 2006.)

[4] Dung, N. V., Hyunh, D. V., Smith P. F. and Wisbauer, R. Extending Modules, PitmanResearch Notes in Mathematics Series (UK: Longman Scientific and Technical, 1994.)

[5] Erdogan, S. E. Absolutely Supplement and Absolutely Complement Modules

(M. Sc. dissertation, Izmir Institute of Technology, 2004) Electronic copy:

http://library.iyte.edu.tr/tezler/master/matematik/T000339.pdf.

[6] Ganesan, L. and Vanaja, N. Modules for which every submodule has a unique coclosure,Comm. Algebra. 30 (5), 2355–2377, 2002.

[7] Generalov, A. I. The ω-cohigh purity in a categories of modules, Math. Notes 33 (5-6)

402–408. Translated from Russian from Mat. Zametki 33 (5), 758–796, 1983.[8] Gerasimov, V. N. and Sakhaev, I. I. A counter example to two hypotheses on projective

and flat modules, Sib. Mat. Zh. 25 (6), 31–35, 1984. English translation: Sib. Math. J. 24,

855–859, 1984.[9] Goodearl, K. R. Ring Theory: Nonsingular Rings and Modules (New York and Basel: Marcel

Dekker Inc., 1976.)

[10] Lam, T. Y. Lectures on Modules and Rings (New York, Berlin, Heidelberg: Springer, 1999.)[11] Mermut, E. Homological Approach to Complements and Supplements (Ph.D. dissertation,

Dokuz Eylul University, 2004.)

[12] Mishina A. P. and Skornyakov, L. A. Abelian groups and modules, American MathematicalSociety Translations Series 2, 107, 1976. Translated from Russian from Abelevy gruppy imoduli, Izdat. Nauka, 1969.

[13] Mohammed, A. and Sandomierski, F. L. Complements in projective modules, J. Algebra127, 206–217, 1989.

[14] Mohamed, S. H. and Muller, B. J. Continuous and Discrete Modules (London Math. Soc.Lecture Notes Series 147, Cambridge, 1990.)


[15] Sklyarenko, E. G. Relative homological algebra in categories of modules, Russian Math.

Surveys 33 (3), 97–137, 1978. Traslated from Russian from Uspehi Mat. Nauk 33 3(201),

85-120, 1978.[16] Talebi, Y. and Vanaja, N. The torsion theory cogenerated by M-small modules, Comm.

Algebra 30 (3), 1449–1460, 2002.

[17] Warfield, R. B. Jr. Serial rings and finitely presented modules, J. Algebra 37, 187–222, 1975.[18] Zoschinger, H. Projektive Moduln mit endlich erzeugtem Radikalfaktormodul, Math. Ann.

255 199–206, 1981.

[19] Zoschinger, H. Schwach-injective moduln, Periodica Mathematica Hungarica 52 (2), 105–128, 2006.


A RESULT ON GENERALIZED DERIVATIONS

IN PRIME RINGS

Yiqiu Du∗ and Yu Wang †‡


Abstract

Let R be a prime ring, H a generalized derivation of R, L a noncentralLie ideal of R, and 0 6= a ∈ R. Suppose that aus(H(u))nut = 0 forall u ∈ L, where s, t ≥ 0 and n > 0 are fixed integers. If s = 0, thenH(x) = bx for all x ∈ R, where b ∈ U , the right Utumi quotient ringof R, with ab = 0 unless R satisfies s4, the standard identity in fourvariables. If s > 0, then H = 0 unless R satisfies s4.

Keywords: prime ring, derivation, generalized derivation, extended centroid, rightUtumi quotient ring.

2000 AMS Classification: 16W25, 16N60, 16R50.

1. Introduction

Throughout this paper, R is always a prime ring with extended centended C, rightUtumi quotient ring U , and two-sided Martindale quotient ring Q. The definitions andproperties of these objects can be found in [3, Chapter 2]. Denote s4 as the standardidentity in four variables.

By a generalized derivation on R one usually means an additive map H : R→ R suchthat H(xy) = H(x)y + xd(y), for some derivation d of R. Obviously any derivation is ageneralized derivation. Another basic example of generalized derivations is the following:H(x) = ax+ xb for a, b ∈ R. Hvala [12] initiated the study of generalized derivations onprime rings. Lee proved the following essential result: every generalized derivation H on adense left ideal of R can be uniquely extended to U and assume the form H(x) = bx+d(x)for some b ∈ U and a derivation d on U [16, Theorem 3]. In recent years, a number ofarticles discussed generalized derivations in the context of prime and semiprime rings(see [1, 5, 9, 10, 11, 18, 19, 21, 22]).

∗Jilin Normal University, College of Mathematics, Siping 136000, China.E-mail:[email protected]†Shanghai Normal University, Department of Mathematics, Shanghai, 200234, China.

E-mail: [email protected]‡Corresponding Author.

82 Y. Du, Y. Wang

Dhara and Sharma [6] proved that, if a ∈ R such that ausd(u)nut = 0 for all u ∈ L,a noncommutative Lie ideal of R, where d a derivation of R, s ≥ 0, t ≥ 0, n ≥ 1 are fixedintegers, then either a = 0 or d = 0 unless char R = 2 and R satisfies s4. Dhara andFilippis [5] proved that, if usH(u)ut = 0 for all u ∈ L, where L a noncommutative Lieideal of R, H a generalized derivation of R, and s, t ≥ 0 are fixed integers, then H = 0unless char R = 2 and R satisfies s4. Recently, the second author [22] investigated thesituation when ausH(u)ut = 0 for all u ∈ L, where L a noncentral Lie ideal of R.

In the present paper we shall generalize the above results in a full general situation.More precisely, we shall prove the following main result of this paper.

1.1. Theorem. Let R be a prime ring, H a generalized derivation of R, L a noncentralLie ideal of R, and 0 6= a ∈ R. Suppose that aus(H(u))nut = 0 for all u ∈ L, wheres, t ≥ 0 and n > 0 are fixed integers. If s = 0, then H(x) = bx for all x ∈ R, where b ∈ Uwith ab = 0 unless R satisfies s4. If s > 0, then H = 0 unless R satisfies s4.

2. The proof of the main result

We begin with the following result, which will be used in the proof of our main result.

2.1. Lemma. Let R be a prime ring with dimCRC > 4. Let 0 6= a ∈ R and b ∈ U suchthat

a[x, y]s(b[x, y])n[x, y]t = 0

for all x, y ∈ R, where s, t ≥ 0 and n > 0 are fixed integers. If s = 0, then ab = 0. Ifs > 0, then b = 0.

Proof. Suppose first that b ∈ C, by assumption we have

abn[x, y]s+n+t = 0

for all x, y ∈ R. It is easy to check that either abn = 0 or R is commutative (see theproof of [17, Theorem 1] or [6, Theorem 2.2]). Hence b = 0 as a 6= 0 and dimCRC > 4.

Suppose next that b 6∈ C. Since R and U satisfy the same generalized polynomialidentity [4, Theorem 2], we have

(2.1) a[x, y]s(b[x, y])n[x, y]t = 0

for all x, y ∈ U . In case C is infinite, the GPI (2.1) is also satisfied by U ⊗C C where Cis the algebraic closure of C. Since both U and U ⊗C C are prime and centrally closed[7], we may replace R by U or U ⊗C C according as C is finite or infinite. Thus we mayassume that R is centrally closed over C which is either finite or algebraically closed suchthat a[x, y]s(b[x, y])n[x, y]t = 0 for all x, y ∈ R.

If s = 0 and ab 6= 0, then a(b[X,Y ])n[X,Y ]t is a nonzero GPI on R as it has nonzeromonomial a(bXY )n(XY )t. By Martindale’s theorem in [20] R is a primitive ring havingnonzero socle and the commuting division D is a finite dimensional central divisionalgebra over C. Since C is either finite or algebraically closed, D must coincide with C.Thus R is isomorphic to a dense subring of EndCV for some vector space V over C. SincedimCRC > 4, it is obvious that dimCV ≥ 3. We will show that, for any given v ∈ V ,v and bv are C-dependent. Assume on the contrary that v and bv are C-independentand set W = Cv + Cbv. Since dimCV ≥ 3, there exists u ∈ V such that v, bv, u are alsoC-independent. If abv 6= 0, by the density of R in EndCV there exist two elements r1and r2 in R such that

r1v = 0, r1bv = 0, r1u = v; r2v = u, r2bv = u, r2u = 0

and so

[r1, r2]v = v and [r1, r2]bv = v.

A Result on Generalized Derivations in Prime Rings 83

Hence,

0 = a(b[r1, r2])n[r1, r2]tv = abv,

a contradiction.Suppose that abv = 0. Since ab 6= 0, there exists w ∈ V such that abw 6= 0 and so

ab(v − w) 6= 0. By the previous argument we have that there exist β, γ ∈ C such that

bw = βw and b(w − v) = γ(w − v).

This yields that (β−γ)w ∈W . Now β = γ implies the contradiction that bv = βv. Thusβ 6= γ and so w ∈ W . But if u ∈ V with abu = 0, then ab(w + u) 6= 0. So w + u ∈ Wforcing u ∈W . Thus V = W and so dimCV = 2, a contradiction.

If s ≥ 1, it is easy to see that a[X,Y ]s(b[X,Y ])n[X,Y ]t is a nonzero GPI on R. By theprevious argument R is isomorphic to a dense subring of EndCV with dimCV ≥ 3. Wewill show that, for any given v ∈ V , v and bv are C-dependent. Assume on the contrarythat v and bv are C-independent and set W = Cv+Cbv. Since dimCV ≥ 3, there existsu ∈ V such that v, bv, u are also C-independent. If av 6= 0, by the density of R in EndCVthere exist two elements r1 and r2 in R such that

r1v = 0, r1bv = 0, r1u = v; r2v = u, r2bv = u, r2u = 0

and so

[r1, r2]v = v and [r1, r2]bv = v.

Hence,

0 = a[r1, r2]s(b[r1, r2])n[r1, r2]tv = av,

a contradiction.Suppose that av = 0. Since a 6= 0, there exists w ∈ V such that aw 6= 0 and so

a(v − w) 6= 0. By the previous argument we have that there exist β, γ ∈ C such that

bw = βw and b(w − v) = γ(w − v).

This yields that (β−γ)w ∈W . Now β = γ implies the contradiction that bv = βv. Thusβ 6= γ and so w ∈ W . But if u ∈ V with au = 0, then a(w + u) 6= 0. So w + u ∈ Wforcing u ∈W . Thus V = W and so dimCV = 2, a contradiction.

Hence, in any case, for all v ∈ V , v and bv are linearly C-dependent. Thus, standardarguments show that b ∈ C, which contradicts our hypothesis. �

We are in a position to give

The proof of Theorem 1.1. We assume that R does not satisfy s4. That is, dimCRC > 4.By a theorem of Lanski and Montgomery [15, Theorem 13] we have 0 6= [I, R] ⊆ L, whereI is a nonzero ideal of R. Hence we may assume without loss of generality that L = [I, I].By [16, Theorem 3] we may assume that H(x) = bx + d(x) for all x ∈ U , where b ∈ Uand d a derivation of U . Thus

a[x1, x2]s(b[x1, x2] + d([x1, x2]))n[x1, x2]t = 0

for all x1, x2 ∈ I. Since I and U satisfy the same differential identities [4], we have

a[x1, x2]s(b[x1, x2] + d([x1, x2]))n[x1, x2]t = 0

for all x1, x2 ∈ U . Assume first that d is Q-inner, i.e., there exists b, c ∈ U such thatH(x) = bx+ xc for all x ∈ U . So

(2.2) f(x1, x2) = a[x1, x2]s(b[x1, x2] + [x1, x2]c)n[x1, x2]t = 0

for all x1, x2 ∈ U . In case C is infinite, the GPI (2.2) is also satisfied by U ⊗C C whereC is the algebraic closure of C. Since both U and U ⊗C C are prime and centrally closed

84 Y. Du, Y. Wang

[7], we may replace R by U or U ⊗C C according as C is finite or infinite. Thus we mayassume that R is centrally closed over C which is either finite or algebraically closed suchthat f(x1, x2) = 0 for all x1, x2 ∈ R.

Suppose first that c 6∈ C. Then f(X1, X2) is a nonzero GPI for R as it has a nonzeromonomial a(X1X2)s(X1X2c)

n(X1X2)t. By Martindale’s theorem in [20] R is a primitivering having nonzero socle and the commuting division D is a finite dimensional centraldivision algebra over C. Since C is either finite or algebraically closed, D must coincidewith C. Thus R is isomorphic to a dense subring of EndCV for some vector space Vover C. Since dimCR > 4, it is obvious that dimCV ≥ 3. We will show that, for anygiven v ∈ V , v and cv are C-dependent. Assume on the contrary that v and cv areC-independent and set W = Cv + Ccv. Since dimCV ≥ 3, there exists u ∈ V such thatv, cv, u are also C-independent. If av 6= 0, by the density of R in EndCV there exist twoelements r1 and r2 in R such that

r1v = 0, r1cv = u, r1u = v and r2v = u, r2cv = 0, r2u = bv − v

and so

[r1, r2]v = v and [r1, r2]cv = −bv + v.

Hence,

0 = a[r1, r2]s(b[r1, r2] + [r1, r2]c)n[r1, r2]tv = av,

a contradiction.Suppose that av = 0. Since a 6= 0, there exists w ∈ V such that aw 6= 0 and so

a(v − w) 6= 0. By the previous argument we have that there exist β, γ ∈ C such that

cw = βw and c(w − v) = γ(w − v).

This yields that (β−γ)w ∈W . Now β = γ implies the contradiction that cv = βv. Thusβ 6= γ and so w ∈ W . But if u ∈ V with au = 0, then a(w + u) 6= 0. So w + u ∈ Wforcing u ∈W . Thus V = W and so dimCV = 2, a contradiction.

Hence, in any case, for all v ∈ V , v and cv are linearly C-dependent. Thus, standardarguments show that c ∈ C which contradicts our hypothesis.

Suppose next that c ∈ C. By our assumption we have

a[x1, x2]s((b+ c)[x1, x2])n[x1, x2]t = 0

for all x1, x2 ∈ U . Then the result follows from Lemma 2.1.Assume next that d is not Q-inner. Then

a[x1, x2]s(b[x1, x2] + [d(x1), x2] + [x1, d(x2)])n[x1, x2]t = 0

for all x1, x2 ∈ U . In view of the powerful Kharchenko’s theorem [14] we have

a[x1, x2]s(b[x1, x2] + [x3, x2] + [x1, x4])[x1, x2]t = 0

for all x1, x2, x3, x4 ∈ U . Setting x3 = ix1 and x4 = 0, where i = 1, 2, we have

(2.3) a[x1, x2]s((b+ i)[x1, x2])n[x1, x2]t = 0

for all x1, x2 ∈ R. If s = 0, we get from Lemma 2.1 that a(b + i) = 0. It follows thata = 0, contradicting our assumption. If s > 0, we get from Lemma 2.1 that b+ i = 0, acontradiction. The proof of the result is complete. �

Acknowledgements

The author would like to express his sincere thanks to the referee for valuable sugges-tions.

A Result on Generalized Derivations in Prime Rings 85

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[14] Kharchenko, V. K. Differential identities of prime rings, Algebra and Logic 17 155-168,1978.

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J. Math. 42, 117-136, 1972.[16] Lee, T. K. Generalized derivations of left faithful rings, Comm. Algebra 27, 4057-4073,

1999.

[17] Lee, T. K. and Lin, J. S. A result on derivations, Proc. Amer. Math. Soc. 124, 1687-1691, 1996.

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29, 4437-4450, 2001.[19] Lin, J. S. and Liu, C. K. Generalized derivations with invertible or nilpotent on multi-

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[22] Wang, Y. Annihilator conditions with generalized derivations in prime rings, Bull.Korean Math. Soc. 48, 917-922, 2011.

STATISTICS


DYNAMICS OF PERFORMANCE AND

TECHNOLOGY IN HIGHER EDUCATION: AN

APPLIED STOCHASTIC MODEL AND

A CASE STUDY

AHMET KARA ∗


Abstract

The purpose of this paper is to develop a stochastic-dynamic model ofperformance and technology in education sector and bring into lightthe presence, in a particular subset of the Turkish higher educationsector, of stochastically-evolving equilibria moving towards a low per-formance trap over time. The dynamics of the movement in questionhinges, in part, on two factors, namely, (1) the productivity growth and(2) student population growth. We formulate a stochastically-driven,technology-based policy option that could help the sector to escapethe trap, moving the sector towards high performance equilibria. Theproposed policy option illustrates that technological transformation ineducational practices could solve a structural problem (a low perfor-mance trap) in developing-country education sectors.

Keywords: Education, universities, low performance trap, economic dimensions, tech-nology, transformation, stochastic-dynamic models.

2000 AMS Classification: 37N40

1. Introduction

Education is unarguably one of the most important forms of investment shaping themodern economies in the twenty-first century. Skills, knowledge and capabilities (i.e.,various dimensions of human capital) acquired or developed through education havebeen among the key determinants of the micro performance of economic actors, institu-tions and sectors as well as the macro performance of contemporary economies. Studies

∗Istanbul Commerce University, Department of Economics, 34672, Istanbul, Turkey.E-mail: [email protected]

88 A. Kara

demonstrating the positive effect of human capital on economic growth have been suffi-ciently indicative of the powerful influence of education on the economies in question [1].†

Studies on education are not, however, limited to the education (or human capital)-induced economic growth. The issues covered by a wide spectrum of works in the litera-ture range from efficient provision of educational services to quality management practicesin higher education. Some of these works, such as [2] and [3] examine issues that cen-ter around funding, equity and efficiency of higher education. Some, such as [8], [12]and [10], analyze the possibility of optimal or strategic educational subsidies. There areother works studying the quality assurance programs and their effects on the demand foreducation [16]. Topics covered by the works in the literature are indeed rich and mul-tidimensional, and yet many dimensions (issues) of this fascinatingly complex area stillremain under-explored. Among these issues is the stochastic economic dynamics in edu-cation (university) sector which we will examine in this paper. We will present an appliedstochastic model of the dynamics in question and demonstrate that a subset of the highereducation sector in Turkey has been trapped into low performance-stochastic-equilibria.We will propose an information technology-driven stochastic policy rule, which helps thesector to get out of such equilibria and reach a stable high performance-target.

In the second section of the paper, we develop the model. The third section presentsthe empirical results. The policy implications are articulated in the fourth section. Theconcluding remarks follow in the fifth section.

2. The Model ‡

Consider an education sector where suppliers (such as universities) provide a service,say x, to the customers. § For the sake of simplicity we will analyze the case of a typicalsupplier in the market. Let Dt denote the quantity demanded for service x supplied bythe firm, which indicates the degree to whichcustomers are willing to buy the serviceat time t. ¶ Dt depends on the relative price of the service at time t (Rt), customers’income at time t (Mt), the service performance at time t (Pt), and the degree to whichinformation technology is used in educational services at time t (Tt).

i.e., Dt = fD (Rt,Mt, Pt, Tt) ,

which is a demand function for the educational service. Rt ∈ (0,∞), Mt ∈ (0,∞).By virtue of the particular way of measuring performance and technology utilization,explained in Section III, Pt and Tt take on values between 1 and 7, i.e., Pt ∈[1,7], andTt ∈ [1, 7]. Dt ∈ (0,∞). Among the variables in the equation, relative price and in-come are standard variables that appear in the conventional specification of demandfunctions. Depending on the empirical case, however, one or both of these variablesmay turn out to be statistically insignificant, rendering their presence in the equation”unnecessary/dispensable”. In the empirical case examined (in the next section) in thispaper, relative price turns out to be such a dispensable variable and is left out of the

†For an example of ”growth-centered” works on human capital, with a different focus, seeTatoglu [18]. There are also works studying the effects of human capital on microeconomicprocesses. An example of such works is Danchev [6], which examines the influence of humancapital/social capital on the sustainable behavior of the firm.‡The model and policy implications benefit from some of the structures presented in Kara

[12].§Educational firms could provide multiple services, in which case x could be conceived as “a

composite service” representing these services.¶This is a ”degree-based” concept of quantity demanded. The concept of quantity supplied

defined below is degree-based as well.

Dynamics of Performance and Technology in Higher Education . . . 89

equation. The other two variables in the equation, namely the use of technology andperformance, are of more specific nature and have contextual relevance to the issue un-der consideration. The use of technology, which is studied in a number of works in theliterature such as Ellis [9] and Goffe and Sosin [11] is key to modern educational practicesand could reasonably affect the demand for educational services. For instance, teachingaccounting software for practical purposes in business and economics departments couldincrease the job performance of the graduates. Same holds for computer-based skillsinvolving statistical, optimization and simulation software. Increased job performanceof graduates increases the demand for graduates in question and hence the demand forhigher education. Similarly, service performance - which is rated on the basis of suchtangible factors as the number of students taught/reached (quantitative teaching perfor-mance), the number of articles published (quantitative research performance), numberof social activities organized by the institution, logistic services (such as dormitory) etc.,and such intangible factors as reliable and effective teaching, responsiveness of personnelto students’ needs etc. – is a potentially relevant variable for the study of demand (aswell as supply, as indicated below) of educational services (Kara [13]).

Let St denote the quantity supplied for the service, which indicates the degree to whichthe supplier is willing to supply the service at time t. Suppose that St depends on therelative price of the service (Rt) as well as on the present and past performances (Pt,Pt−1), i.e., the supply function for the educational service is:

St = fS(Rt, Pt, Pt−1).‖

Rt ∈ (0,∞), Pt ∈ [1, 7], Pt−1 ∈ [1, 7], and St ∈ (0,∞).For analytical purposes, we will assume that the demand and supply functions have

the following explicit (log-linear) forms: ∗∗

lnDt = α0 + α1lnPt + α2lnMt + α3lnRt + α4lnTt + ut

and

lnSt = β0 + β1 lnPt + β2 lnPt−1 + β3 lnRt + vt

where ut and vt are independent normally distributed white noise stochastic terms uncor-related over time. They have zero means and constant variances σ2

u and σ2v respectively.

Here a specific feature of the service supply of the higher education institutions inTurkey needs to be noted: Even at low performance levels, many of these institutionsdo end up supplying services. The level of these services at time t depends on the levelof these services at t = 0, and the growth rate of these services reflecting roughly thegrowth of student population in the system. Let, in the absence of stochastic shocks,and at the minimal performance levels, St have the value of A, which grows at a rate ofg over time. Thus, at Pt = 1 and Pt−1 = 1, St = A(1 + g)t ⇒ lnSt = t.lnA(1 + g) = β0(By the argument presented in the subsection on supply behavior below, the effects ofprices have been left out).

To theorize about the movements over time (i.e., the dynamic trajectory) of serviceperformance, we will make two reasonable assumptions: First, the relative strength (ormagnitude) of the demand compared to the supply provides an impetus for performanceto be adjusted upwards over time. Second, productivity growth contributes to perfor-mance improvements over time. These assumptions are relevant to the Turkish educa-tional system in the following respects. Regarding the first assumption, in Turkey, thegap between the demand for higher educational services and the supply has been a keysource of pressure for the increased volume of higher educational services, which is a key

∗∗Needless to say, of course, that log linear forms are extensively used throughout economicliterature.

90 A. Kara

determinant of service performance. The wider is the gap between demand and supply,the stronger is the pressure on higher education institutions to increase services. Thus,it is reasonable to assume that the relative strength of the demand compared to supplyprovides an impetus for performance to be adjusted upwards over time. To exemplify therelevance of the second assumption, suppose, for instance, that due to asupplydvancesin information technology (such as e-learning), universities are able to reach a highernumber of students with the same number of teachers. This is an increase in the averageproductivity of teachers, which, of course, increases the overall teaching performance ofthe universities.

Taking these factors into account, we formulate the following adjustment dynamic forperformance.

Pt+1/Pt = (Dt/St)k (1 + δ)t ,

where k is the coefficient of adjustment and δ is a productivity growth at t.Taking the logarithmic transformation of both sides, we get:

lnPt+1 = lnPt + k (lnDt − lnSt) + t (ln (1 + δ))

We will call this the dynamic adjustment equation. Substituting the functional ex-pressions (forms) for lnDt and lnSt specified above, setting the values of Mt, Rt and Ttto their average values Mavr, Ravr and T avr and rearranging the terms in the equation,we get,

lnPt+1+(kβ1 − kα1 − 1) lnPt+kβ2 lnPt−1 = k(α0+α2 lnMavr+(α3−β3)lnRavr

+ α4lnTavr) + k (ut − vt)− k [lnA(1 + g)] t+ [ln (1 + δ)] t,

which is a second order stochastic difference equation, the solution of which is providedin Appendix A.

The solution in Appendix A shows that the intertemporal equilibrium performance,P ∗ is:

P ∗ = exp

{k (α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

k (β1 + β2 − α1)

+(k lnA(1 + g)− ln (1 + δ)) (1− kβ2)

(k (β1 + β2 − α1))2− k lnA(1 + g)− ln (1 + δ)

k (β1 + β2 − α1)t

+λ1

λ1 − λ2

∞∑j=0

λj1zt−j +λ2

λ2 − λ1

∞∑j=0

λj2zt−j

}where zt = k(ut − vt)

λ1λ2 = kβ2

λ1 + λ2 = 1− kβ1 + kα1

In case where λ1 and λ2 are conjugate complex numbers, i.e., λ1, λ2 = h ∓ vi =r (cos θ ∓ i sin θ) , the intertemporal equilibrium performance is:

P ∗ = exp

{k (α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

k (β1 + β2 − α1)

+(k lnA(1 + g)− ln (1 + δ)) (1− kβ2)

(k (β1 + β2 − α1))2− k lnA(1 + g)− ln (1 + δ)

k (β1 + β2 − α1)t

+

∞∑j=0

rjsin θ(j + 1)

sin θzt−j

}where r is the absolute value of the complex number, and sinθ = v/r and cosθ = h/r.


To study whether this intertemporal equilibrium performance is high or low, andwhether it remains stable over time, we need to empirically estimate the parametersinvolved. This is done in the next section.

3. Empirical Analysis

3.1. The sample. Data for this study was gathered using a questionnaire includingquestions about demand, supply, income, prices, performance and the use technology inthe educational services in Turkey. 100 respondents were asked to answer the questions.66 useable questionnaires were returned giving a response rate of 66 percent, which wasconsidered satisfactory for the analysis in the paper. Some responses with considerablemissing information were excluded. Each question (item) was rated on a seven-pointLikert scale with 1 representing the lowest score that can be assigned, and 7 representingthe highest. (The reason for choosing a seven-point scale is simple. In the literature,researchers often use a five-point scale or a seven-point scale in the questionnaires. I havechosen to use a seven-point scale in order to capture the differences between the answersto the questions in a more sensitive manner)[12].

The information in the questionnaire is used to estimate the parameters in the re-gression equations in the following manner: Each variable in the model is represented bya question in the questionnaire. Thus responses to questions will be the values for thevariables. For instance, Tt represents the degree to which technology is used in educa-tional practices - with 1 representing the lowest use and 7 the highest. Values between1 and 7 represent varying degrees to which the developments in technology (such as ininformation technology) are put in practice in classroom instruction in particular, andknowledge, capacity and skill formation in general. Mt represents customer incomes,which are translated into bands. Bands are, in turn, rated on a seven point scale, with1 representing the lowest income interval and 7 representing the highest income interval.Other questions (variables) are directly rated on a seven point scale. Thus, our samplethat consists of answers to the questions in the questionnaire contains integer values (from1 to 7) for the variables associated with demand, performance, income, use of technologyetc. In order to run a regression relating to, for instance, demand, we regress quantitydemanded for the educational service on performance, income, the use of technology etc.

3.2. Estimation of the parameters. To estimate the parameters involved, we makeuse of the demand and supply equations for the educational service, i.e., equations (4)and (5).

3.2.1. Demand. In the particular empirical case under consideration, public provision ofthe educational service could be considered free, and the differences between the relativeprices of private educational institutions in the sample are insignificant, thus relativeprices do not appear to play a deciding role in the demand for the educational servicein question, therefore the relative price variable is left out of the demand equation. Theregression-results are as follows:

lnDt = −0.756 + 0.428lnPt + 0.358lnMt + 0.598lnTt

(−2.008) (1.827) (2.195) (2.939)

92 A. Kara

R2 = 0.58. t-statistics are given in the parentheses. Thus,

α0 = −0.756

α1 = 0.428

α2 = 0.358

α3 = 0

α4 = 0.598

3.2.2. Supply. Supply is largely determined by central bureaucratic authorities whosedecisions are based on certain criteria, such as the adequacy and quality of physicalinfrastructure and human resources, rather than prices. Thus prices could be convenientlyleft out of the supply function. Rt drops out of the log-linear formulation of the supplyequation. To estimate the other parameters of the supply equation, we asked officials ofthe relevant institutions questions, the answers of which were designed to give the valuesof the elasticities of supply with respect to the present and past d performances. Theanswers indicate that a 1% increase in the past performance would increase the quantitysupplied by about 0.25 %, but a 1% increase in the present performance would increasethe quantity supplied by about 0.75%. However, by virtue of the enrollment constraintsplaced by the Higher Education Council, what the institutions under examination couldsupply was 90 % of what they were willing to supply, Thus,

β1 = 0.9 · 0.75 = 0.675

β2 = 0.9 · 0.25 = 0.225

β3 = 0

The value of A is normalized to 1.

3.2.3. The coefficient of adjustment (k). For simplicity, we will assume that Pt+1/Pt isproportional to the ratio of demand to supply, and hence, k = 1.

Given the empirical values of the parameters obtained above, we get,

λ1 = 0.376 + 0.288i and λ2 = 0.376− 0.288i

We will now consider a particularly interesting case where the student populationgrowth is equal to the productivity growth, i.e., g = δ. This special case is interesting andworthwhile to consider because the equality between the growth rates in question ensuresthe sustainable provision of the educational service. Student population growth ratesroughly represent ”increases in the demand for the educational service” while productivitygrowth rates represent ”increases in the supply (provision) of the educational service”.Starting from an equilibrium, equal demand and supply growth rates enable supply tomeet demand, in a sustainable manner, over time.

With this assumption and with all the needed parameter values at hand, the intertem-poral equilibrium performance is [12]:

P ∗ = exp

{1.29 +

∞∑j=0

0.47jsin θ(j + 1)

sin θzt−j

}

For analytical convenience, we will carry out some of our analysis in terms of loga-rithmically transformed performance, lnP ∗, rather than P ∗. Since function is an order-preserving transformation, analysis in terms of lnP ∗ and P ∗ will yield the same quali-tative results; and the quantitative results could be transformed into one another. The


expected value of the logarithmically transformed intertemporal equilibrium performanceis:

E( lnP ∗) = 1.29 +

∞∑j=0

0.47jsin θ(j + 1)

sin θE(zt−j)

Since, by virtue of the assumptions about ut, and vt, E(ut) = 0, and E(vt) = 0,E(zt) = E(ut)− E(vt) = 0. Thus,

E( lnP ∗) = 1.29

In view of the logarithmically transformed performance scale of ln1 = 0 to ln7 ∼= 1.95,an intertemporal equilibrium expected performance of 1.29 is low (or, at best, mediocre).As proven in Appendix A, this low performance is also stable over time in the particularsense that it has a stationary distribution with a constant mean and variance. Thisindicates a low-performance trap facing the sector across time.††

To elaborate on the stability property of the equilibrium obtained, we will present ageneral description of the stability conditions in terms of k, α1, β1, β2, and then focuson the specific case that is relevant to the numerical values of the parameters estimatedabove. Consider the three different cases concerning the roots of the complementaryfunction associated with the stochastic difference equation (8), i.e.,

lnPt+1 + (kβ1 − kα1 − 1) lnPt + kβ2lnPt−1 = 0.

Stability conditions for the three different cases are as follows:(i) The case of distinct real roots.

Stability conditions:∣∣∣∣∣∣− (kβ1 − kα1 − 1) +((kβ1 − kα1 − 1)2 − 4kβ2

) 12

2

∣∣∣∣∣∣ < 1

and ∣∣∣∣∣∣− (kβ1 − kα1 − 1)−((kβ1 − kα1 − 1)2 − 4kβ2

) 12

2

∣∣∣∣∣∣ < 1

(ii) The case of repeated real roots.The stability condition:∣∣∣∣− (kβ1 − kα1 − 1)

2

∣∣∣∣ < 1

(iii) The case of complex roots.Stability condition:

0 < R = (kβ2)12 < 1.

Since the parameter estimates we obtained above lead to complex roots, the case relevantto out work is (iii), which requires that the square root of kβ2, hence kβ2 itself, be lessthan 1. Since k =1, β2 < 1. What is the meaning and implication of this condition? β2is nothing but the elasticity of supply with respect to past performance. The conditionrequiring that β2<1 implies that supply of educational services should not be “too sen-sitive” to past performances. More explicitly, a 1% increase in past performance should,

††It is possible to formulate the problems in terms of a concept of quality as well. An exampleof such formulation in the context of a different sector is presented by Kara and Kurtulmus

[14]. There are other studies that deal with the quality issues differently. Among the works

that explore the issues of quality in education, for instance, from a “total quality management”perspective are Dahlgaard, Kristensen and Kanji [4, 5].

94 A. Kara

in the future, lead to a less than 1 % increase in the quantity supplied of educationalservices. The condition, in a way, limits the effect of the “past” on the “present”. In theabsence of such a condition, the educational sector under examination would exhibit adestabilizing dynamic similar to the ones observed in some agricultural sectors.

Since our estimate of β2 is 0.225, the condition stipulated in (iii) is met. The lowperformance equilibrium is deterministically stable. It is also stochastically stable in theparticular sense mentioned (and proved) in Appendix A.

The following section will formulate a stochastic policy rule, which will enable the sec-tor to escape the low performance equilibrium by helping the sector to reach a performance-target, and which will stabilize the sector around that target.

4. Policy Implications ‡‡: An Example

Suppose that the educational service providers in Turkey aim to reach a stable (sus-tainable) performance-target in the presence of stochastic shocks. Consider a stochasticshock to demand, in the magnitude of ut, which may have come, for instance, from asudden reduction in the demand for education induced by the recent economic slowdown.Let us design the following demand-side stochastic policy response (rule):

R = η0 + η1ut

where η0 and η1 represent the non-stochastic and stochastic components respectively.Suppose that the non-stochastic component will take the form of a governmental supportfor the use of technology in education. Let η0 = ln ∆T . The government will providesupport so as to increase the use of technology in education by ∆T.

Reaching a stable (”minimally varying”) expected (logarithmically expressed) perfor-mance target in the presence of stochastic shocks turns out to be equivalent to minimizingthe expected loss function of the following kind at the intertemporal equilibrium:

E[(lnPt− lnP

∗∗)2],

where lnP ∗∗ is the (logarithmically expressed) performance target. Decomposing theexpected loss function , we get,

E[(lnPt− lnP ∗∗)

2]

= E[((lnPt− E (lnPt)) + (E (lnPt)− lnP ∗∗))

2]

= E((lnPt− E (lnPt))2

)+ E

((E (lnPt)− lnP ∗∗)

2)

+ 2E (lnPt− E (lnPt)) (E (lnPt)− lnP ∗∗)

Since (E (lnPt)− lnP ∗∗) is not random and since E(lnP t − E(lnP t)) = E(lnP t) −E(lnP t) = 0, the decomposition will take the form of:

E[(lnPt− lnP ∗∗)

2]

= E((lnPt− E (lnPt))2

)+ (E (lnPt)− lnP ∗∗)

2.

The first term represents the variance of (logarithmically expressed) performanceand the second term denotes the ”squared deviation” around lnP ∗∗. Thus minimiz-ing expected loss is equivalent to minimizing the squared deviation, which requires thatexpected (logarithmically expressed) performance be equal to the (logarithmically ex-pressed) performance target, and minimizing the variance of (logarithmically expressed)performance, enabling the educational service provider to reach a stable (minimally vary-ing) logarithmically expressed performance target [12].

‡‡In the literature, there are a number of works exploring various policy issues which rangefrom comprehensive school reform (Desimone [7]) to cost reduction strategies in higher education

(Leonard [15]).For a similar decomposition, though in a different context, see Sargent [17].


To find, for the special case where g = δ, the parameters of the stochastic policy rulewhich minimize the expected loss function, let us incorporate the rule into the function.

E[(lnPt− lnP ∗∗)2

]=

{∞∑j=0

0.472j(

sin θ(j+1)sin θ

)2}((1 + η1)σ2

u + σ2v

)+

{α0 + α2 lnMavr + α4 lnT avr

β1 + β2 − α1+

η0β1 + β2 − α1

− P ∗∗}2

The values of η0 and η1 that minimize the expected loss function are:

η0 =

{P ∗∗ − α0 + α2 lnMavr + α4 lnT avr

β1 + β2 − α1

}(β1 + β2 − α1)

η1 = −1.

For instance, for lnP ∗∗ = 1.5, the value of η0 is calculated to be 0.099. This impliesthat ∆T = 1.1, which represents the required policy-induced increase in the use oftechnology for the performance target in question. The fact that η1 = −1 implies that,for stabilization against the kind of demand shock exemplified here, demand should beincreased by the magnitude of the stochastic shock.

In sum, the designed policy response requires an increase in the use of technology atthe intertemporal equilibrium, captured by the non-stochastic component of the policy,and an increase in the stochastic component so as to meet the temporary reduction inthe demand for services.

5. Concluding Remarks

The paper develops an applied stochastic model of higher education sector in Turkey,and shows that the sector could, under certain conditions, slide into a low-performancetrap over time. The paper presents a stochastic policy option that could help thesector to avoid the trap in question. The designed stochastic resolution and the model,however, focus on the overall performance of the educational institutions, and as such,do not take into account the micro components of the overall performance. Decomposingthe overall performance into its micro components could more precisely reveal the sourcesof potential improvements in overall performance, opening up new possibilities for policyformulations. The decomposition in question could be done in a number of ways. Forinstance, a functional decomposition of the overall performance into the componentsof “teaching performance” and “research performance” would enable the institutions toexactly identify their teaching-related or research-related strengths and weaknesses, andhence would help them to take concrete/measurable steps towards their teaching-and-research-related-targets. Alternatively, the overall performance could be decomposedinto its departmental components, identifying the contributions of each department tothe overall performance. Dynamic modeling based on such decomposition could providekey information about the strategic decisions about how departmental priorities shouldbe arranged and revised over time.

Finally, we will elaborate on one feature of the paper that can be relaxed, namelythat supply and demand functions are of log-linear form. There are two main reasonsfor choosing the log-linear form for the functions used in the paper. First, the log-linearform is one of the most extensively used forms in the economic literature, which madeit the prime candidate for our work as well. Second, the log-linear form has a peculiarfeature that the coefficients of the variables represent the “elasticities” of the dependentvariable with respect to the relevant independent variables. This has made it possible todirectly estimate the parameters of the supply function in Section 3.

A similar trap in the banking sector is studied in the literature (e.g., Kara and Kurtulmus [14]).

96 A. Kara

Do we obtain similar results when the supply and demand functions are nonlinear?Our conjecture is that, with certain types of nonlinear functions, the central result con-cerning the low performance trap could still be obtained. For instance, in certain equi-librium models with particularly specified quadratic functions, a similar result could beobtained through computer-based simulations. However, in view of the wide range ofpossible nonlinear functions, it would be difficult to make general definitive statementsfor all cases where functions are nonlinear. Nonlinearity could, in some cases, lead tomultiple equilibria with different stability properties. Such cases, which may not lendthemselves to analytical solutions and which could be studied through computer-basedsimulations or other means, are worthy of future research.

6. Appendix A

The solution for the second order stochastic difference equation,

lnPt+1 + (kβ1 − kα1 − 1)lnPt + kβ2lnPt−1

= k(α0 + α2lnMavr + (α3 − β3)lnRavr + α4lnT

avr)

+ k(ut − vt)− [klnA(1 + g)]t+ [(ln(1 + δ)]t,

has two components, namely a particular solution and a complementary function. Wewill find these components for lnPt and then take the anti-log of lnPt so as to find thesolution for Pt.

(1) Particular Solution: Letting xt = lnPt, using the lag operator L (defined asLiPt = Pt−i, for i = 1, 2, 3. . . ), and rearranging the terms, we get the followingform of the second order stochastic difference equation above,

[1− (1− kβ1 + kα1)L− (−kβ2)L2]xt = k(α0 + α2lnMavr + (α3 − β3)lnRavr

+ α4lnTavr) + k(ut − vt)− [klnA(1 + g)]t+ [(ln(1 + δ)]t,

which could be transformed into,

(1− λ1L)(1− λ2L)xt = k(α0 + α2lnMavr + (α3 − β3)lnRavr + α4lnT

avr)

+ k(ut − vt) − [klnA(1 + g)]t + [(ln(1 + δ)]t,

where

λ1λ2 = kβ2

λ1 + λ2 = 1− kβ1 + kα1

Thus, we get,

xt = (1− λ1L)−1(1− λ2L)−1[k(α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

+ k(ut − vt)− [k lnA(1 + g)− ln(1 + δ)]t]

Using the properties of partial fractions,

(1− λ1L)−1(1− λ2L)−1 =λ1

λ1 − λ2(1− λ1L)−1 +

λ2

λ2 − λ1(1− λ2L)−1

Thus,

xt =λ1

λ1 − λ2(1− λ1L)−1 +

λ2

λ2 − λ1(1− λ2L)−1

[k(α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

+ k(ut − vt)− [k lnA(1 + g)− ln(1 + δ)]t]


Using the properties of some series and lag operators, and doing some algebraicmanipulations, we get,

xt =k(α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

k(β1 + β2 − α1)

+(k lnA(1 + g)− ln(1 + δ))(1− kβ2)

(k(β1 + β2 − α1))2− k lnA(1 + g)− ln(1 + δ)

k(β1 + β2 − α1)t

+λ1

λ1 − λ2

∞∑j=0

λj1zt−j +λ2

λ2 − λ1

∞∑j=0

λj2zt−j

where zt = k(ut − vt)This is the parametric expression of xt = lnPt, at the intertemporal equilib-

rium. Let P∗ denote the intertemporal equilibrium performance. Thus,

P ∗ = exp[k(α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

k(β1 + β2 − α1)

+(k lnA(1 + g)− ln(1 + δ))(1− kβ2)

(k(β1 + β2 − α1))2− k lnA(1 + g)− ln(1 + δ)

k(β1 + β2 − α1)t

+λ1

λ1 − λ2

∞∑j=0

λj1zt−j +λ2

λ2 − λ1

∞∑j=0

λj2zt−j ]

In case where λ1 and λ2 are conjugate complex numbers, i.e., λ1, λ2 = h±vi =r(cosθ ± isinθ), the intertemporal equilibrium performance is:

P ∗ = exp[k(α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

k(β1 + β2 − α1)

+(k lnA(1 + g)− ln(1 + δ))(1− kβ2)

(k(β1 + β2 − α1))2− k lnA(1 + g)− ln(1 + δ)

k(β1 + β2 − α1)t

+

∞∑j=0

rjsin θ(j + 1)

sin θzt−j ]

where r is the absolute value of the complex number, and sinθ = v/r andcosθ = h/r.

(2) Complementary Function: To find this component of the solution, we needto consider the following reduced form of the second order difference equation.

lnPt+1 + (kβ1 − kα1 − 1)lnPt + kβ2lnPt−1 = 0.

A possible general solution could take the form lnPt = Ayt. Through thestandard procedure, we obtain,

y2 + (kβ1 − kα1 − 1)y + k(β2) = 0.

This quadratic equation could have at most two roots. Suppose that k = 1.Substituting α1 = 0.428, β1 = 0.675 and β2 = 0.225 into the quadratic equationand solving it for the roots, we get,

y1 = 0.376 + 0.288i

and

y2 = 0.376− 0.288i.

Thus, the solution for the reduced equation is

A1yt1 +A2y

t2 = A1(0.376 + 0.288i)t +A2(0.376− 0.288i)t

98 A. Kara

where A1 and A2 are non-zero constants. This could be shown to be equivalentto:

0.47t(A3cosθt+A4sinθt)

where A3 = A1 + A2 and A4 = (A1 − A2)i. sinθ = 0.288/0.47 and cosθ =0.376/0.47.

(3) The general solution: The general solution for the equation is the sum of thetwo solutions obtained in (a) and (b),

lnP ∗ = [k(α0 + α2 lnMavr + (α3 − β3) lnRavr + α4 lnT avr)

k(β1 + β2 − α1)

+(k lnA(1 + g)− ln(1 + δ))(1− kβ2)

(k(β1 + β2 − α1))2− k lnA(1 + g)− ln(1 + δ)

k(β1 + β2 − α1)t

+

∞∑j=0

rjsin θ(j + 1)

sin θzt−j ] + 0.47t(A3 cos θt+A4 sin θt).

In the paper, we analyze the case where g = δ. Substituting the values of the param-eters involved, we get, for this special case,

lnP ∗ = 1.29 +

∞∑j=0

0.47jsin θ(j + 1)

sin θzt−j + 0.47t (A3 cos θt+A4 sin θt)

The values of A3 and A4 could be obtained by specifying two initial conditions. How-ever, for the purposes of our analysis, we do not need to know the values of those con-stants. Since the absolute value of the complex number involved is 0.47, which is lessthan 1, as t → ∞, 0.47t(A3cosθt + A4sinθt) will converge toward zero, and hence thegeneral solution converges toward the particular solution,

lnP ∗ = 1.29 +

∞∑j=0

0.47jsin θ(j + 1)

sin θzt−j

Thus,

E (lnP ∗) = 1.29 +

∞∑j=0

0.47jsin θ(j + 1)

sin θE (zt−j)

Since, by virtue of the assumptions about ut, and vt, E(ut) = 0, and E(vt) = 0,E(zt) = k(E(ut)− E(vt)) = 0. Thus,

E(lnP ∗) = 1.29.

which is nothing but the intertemporal expected equilibrium performance. Note that ut,and vt are uncorrelated over time, and so is zt. They have zero covariances. Thus, thevariance (V ) of lnP ∗ is,

V (lnP ∗) = V

{1.29 +

∞∑j=0

0.47jsin θ(j + 1)

sin θzt−j

}

V (lnP ∗) =

{∞∑j=0

0.472j

(sin θ(j + 1)

sin θ

)2}(

σ2u + σ2

v

)which is constant. (Please note the value of sinθ specified above). Needless to say,taking the limits of mean and variance as t→∞ yields the same results. For the sake ofexposition,

limt→∞

E (lnP ∗) = 1.29,


and

limt→∞

V (lnP ∗) =

{∞∑j=0

0.472j

(sin θ(j + 1)

sin θ

)2}(

σ2u + σ2

v

)Thus, logarithmically transformed intertemporal performance has a stationary distribu-tion in the sense that it has a constant mean and variance.

References

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[3 ] Hurvich, C. M. and Tsai, C. L. Regression and time series model selection

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