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““DERIVATIONS AND DERIVATIONS AND RELATED MAPS” RELATED MAPS”

byby

Dr. M. S. SammanDr. M. S. Samman

Dr. A. B. ThaheemDr. A. B. Thaheem

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ContentsContents

1.1. IntroductionIntroduction

2.2. Definitions and notationsDefinitions and notations

3.3. Background of the problem(s)Background of the problem(s)

4.4. ObjectivesObjectives

5.5. ResultsResults

6.6. ReferencesReferences

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1. Introduction1. Introduction The theory of derivations plays an important role The theory of derivations plays an important role

in quantum physics. The study of bounded in quantum physics. The study of bounded derivations on operator algebras started in 1950s derivations on operator algebras started in 1950s and has been very actively carried out by many and has been very actively carried out by many mathematicians, and it has become a useful mathematicians, and it has become a useful mathematical theory. After the study of bounded mathematical theory. After the study of bounded derivations (see Kadison [26], Sakai [37, 38] ), the derivations (see Kadison [26], Sakai [37, 38] ), the study of unbounded derivations is being carried on study of unbounded derivations is being carried on by many mathematicians and mathematical by many mathematicians and mathematical phycisists, and this theory is making great progress phycisists, and this theory is making great progress (see Bratteli and Robinson [6]). (see Bratteli and Robinson [6]).

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1. Introduction1. Introduction (cont’d) (cont’d) Derivations have also been studied for Banach algebras with Derivations have also been studied for Banach algebras with

particular attention to the problem of automatic continuity particular attention to the problem of automatic continuity of derivations with reference to the Singer-Wermer of derivations with reference to the Singer-Wermer conjecture [39] which has bene finally resolved by Thomas conjecture [39] which has bene finally resolved by Thomas [47] in 1988.[47] in 1988.

   C*-algebras are semiprime rings and factors (von Neumann C*-algebras are semiprime rings and factors (von Neumann

algebras with center consisting of scalar multiples of the algebras with center consisting of scalar multiples of the identity operator) are prime rings. Therefore, it is natural identity operator) are prime rings. Therefore, it is natural to consider derivations on prime and semiprime rings.to consider derivations on prime and semiprime rings.

Posner [36] initiated several aspects of a study of derivations Posner [36] initiated several aspects of a study of derivations such as composition of derivations, commuting and such as composition of derivations, commuting and centralizing derivations on prime rings.centralizing derivations on prime rings.

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1. Introduction1. Introduction (cont’d) (cont’d) Posner [36], proved that zero is the only centralizing Posner [36], proved that zero is the only centralizing

derivation on a non-commutative prime ring. J. Mayne [32, derivation on a non-commutative prime ring. J. Mayne [32, 33] proved the analogous result for centralizing 33] proved the analogous result for centralizing automorphisms on prime rings. A number of authors have automorphisms on prime rings. A number of authors have generalized these theorems of Posner and Mayne in several generalized these theorems of Posner and Mayne in several ways (see e.g. [3, 7, 10, 12, 27, 29, 48, 49]). Further, a lot of ways (see e.g. [3, 7, 10, 12, 27, 29, 48, 49]). Further, a lot of work has been done on the general theory of derivations and work has been done on the general theory of derivations and related generalized notions on prime and semiprime rings related generalized notions on prime and semiprime rings (see e.g. [4, 8, 9, 18, 23-25]).(see e.g. [4, 8, 9, 18, 23-25]).

Derivations have also been extended to Derivations have also been extended to -derivations-derivations and ( and (, , ββ)-derivations where )-derivations where and and ββ are automorphisms. These are automorphisms. These derivations are also called skew-derivations. (see [19, 28, 44]).derivations are also called skew-derivations. (see [19, 28, 44]).

  

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1. Introduction1. Introduction (cont’d) (cont’d) This work is concerned with the general theory of This work is concerned with the general theory of

derivations and related maps on prime and semiprime derivations and related maps on prime and semiprime rings. We shall summarize some results obtained in this rings. We shall summarize some results obtained in this direction. In fact, our study here deals with general direction. In fact, our study here deals with general properties of derivations, properties of derivations, -derivations, centralizing and -derivations, centralizing and commuting maps, a functional equation and commuting maps, a functional equation and automorphisms for prime and semiprime rings. Any automorphisms for prime and semiprime rings. Any extension of these and other results to near-rings remains extension of these and other results to near-rings remains an open problem.an open problem.

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Definitions & NotationsDefinitions & Notations

R R denotes a ring with center Z(denotes a ring with center Z(RR))

[[x, yx, y] ] = xy - yx = xy - yx forfor x, y x, y inin R. R.

[[xy, zxy, z]] = x = x[[y, zy, z]] + + [[x, zx, z]]y & y & [[x, yzx, yz] ] = y= y[[x, zx, z] ] + + [[x, yx, y]]z z

forfor x, y, z x, y, z R.R.

R R isis primeprime if if aRb = aRb = (0)(0) impliesimplies a= a=00 or b= or b=00

RR is is semiprimesemiprime if aRa = if aRa = (0) implies(0) implies a = a = 0 0

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Definitions & Notations (cont’d)Definitions & Notations (cont’d)

An additive mapAn additive map d: R d: R→→ R R is called ais called a derivation derivation if if d(x y) = dd(x y) = d((xx))y + xdy + xd((yy)) for allfor all x, y x, y R.R.

A A mappingmapping f : R f : R→→ R R is calledis called centralizing centralizing if if

[[f(x), xf(x), x]] Z Z((RR)); ; in particular, ifin particular, if [[ff((xx)), x, x]] = = 00

for allfor all x x R, R, then it is calledthen it is called commutingcommuting..

It is easy to see that if It is easy to see that if f: R→Rf: R→R is additive and is additive and

commuting map then [commuting map then [ff((xx)), y, y] = [] = [x, fx, f((yy)] )] x, yx, y RR..

Clearly, Clearly, commuting commuting →→ centralizing centralizing

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Definitions & Notations (cont’d)Definitions & Notations (cont’d)

An additive mapAn additive map f f on a ringon a ring R R is calledis called antihomomorphism antihomomorphism if if ff((xyxy)) = f = f((yy))ff((xx) ) for allfor all x, x, y y R.R.

A mappingA mapping f: R→R f: R→R is calledis called commutativity- commutativity- preservingpreserving if [if [ff((xx)), f, f((yy)])] = = 00 wheneverwhenever [[x, yx, y] = 0] = 0..

A mapping A mapping f: R→R is f: R→R is calledcalled strong strong commutativity-preservingcommutativity-preserving if [if [ff((xx)), f, f((yy)])] = = [[x, yx, y]] for allfor all x, y x, y R. R.   

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Various aspects of the derivation are in Various aspects of the derivation are in relations with operators algebras, C*-algebras, relations with operators algebras, C*-algebras, von Neumann algebras, etc.von Neumann algebras, etc.

Background of the problem(s)Background of the problem(s)

The use of derivations in operators equations.The use of derivations in operators equations.

Centralizing automorphisms have impact on Centralizing automorphisms have impact on the theory of C*-algebra.the theory of C*-algebra.

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4. Objectives4. Objectives

VI.VI. Derivations and related mappings in near- Derivations and related mappings in near- rings (?)rings (?)

I.I. General properties of derivationsGeneral properties of derivations

II.II. -derivations on prime and semiprime rings-derivations on prime and semiprime rings

III.III. Commuting and centralizing derivationsCommuting and centralizing derivations

IV.IV. Commuting and centralizing automorphismsCommuting and centralizing automorphisms

V.V. Certain functional equations on prime and Certain functional equations on prime and semiprime rings semiprime rings

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5. Results5. Results

Results -AResults -A Related to Results -ARelated to Results -A

Lemma A1Lemma A1: :

Let Let RR be a semiprime ring, be a semiprime ring, II a nonzero two-sided ideal of a nonzero two-sided ideal of RR and and aa R such that R such that axaaxa = 0 for all = 0 for all xx II, then , then a = a = 0.0.

Theorem A2Theorem A2 : :

Let Let RR be a semiprime ring, be a semiprime ring, II a nonzero two-sided ideal of a nonzero two-sided ideal of RR and and f, gf, g be derivations of be derivations of RR such that such that ff((xx))yy + + ygyg((xx) = 0 for all ) = 0 for all x, yx, y I I. Then . Then ff((uu) [) [x, yx, y] = [] = [x,yx,y]]gg((uu) = 0 for all ) = 0 for all x ,y ,ux ,y ,u II; in ; in particular, particular, f f and and gg map map I I into into ZZ((RR). ).

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5. Results (cont’d) -5. Results (cont’d) -AA

Corollary A3Corollary A3 : :

Let Let RR be a noncommutative prime ring, be a noncommutative prime ring, II a nonzero two- a nonzero two-sided ideal of sided ideal of RR and and f, gf, g be derivations of be derivations of RR such that such that

ff((xx)) y y + y + ygg((xx) = 0 for all ) = 0 for all x, yx, y I I. Then . Then f = g = f = g = 0 on 0 on RR. . Remark A4Remark A4 : :

If If RR is a noncommutative prime ring, is a noncommutative prime ring, I I a nonzero two-sided ideal a nonzero two-sided ideal of of RR such that such that f(x)xf(x)x = 0 for all = 0 for all xx II, then , then ff = 0 on = 0 on RR. This follows . This follows from Corollary A3. Indeed, put from Corollary A3. Indeed, put gg = 0 and = 0 and y=xy=x in (A3), we get in (A3), we get f(x)xf(x)x = 0 for all = 0 for all xx I I and hence and hence f f = 0 on = 0 on RR. .

Theorem A5Theorem A5 : :

Let Let RR be a noncommutative prime ring, be a noncommutative prime ring, I I a nonzero two-sided a nonzero two-sided ideal of ideal of RR and and f, gf, g be derivations of be derivations of R R such that such that

ff((xx))xyxy + + ygyg((xx))xx = 0 for all = 0 for all x, yx, y II. Then. Then f f = = gg = 0. = 0.

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5. Results (cont’d)5. Results (cont’d) Results -BResults -B

Related to Results -BRelated to Results -B Theorem B1Theorem B1: :

Let Let TT be an endomorphism of a 2-torsion free semiprime ring be an endomorphism of a 2-torsion free semiprime ring RR such that the mapping x [such that the mapping x [TT((xx)), x, x] is ] is commutingcommuting on on RR. Then . Then ([([TT((xx)), x, x])])22 =0 for all =0 for all x x RR . .

Corollary B2Corollary B2: :

Let Let TT be a be a centralizingcentralizing endomorphism of a 2-torsion free endomorphism of a 2-torsion free semiprime ring semiprime ring RR, then it is commuting on , then it is commuting on RR. .

Theorem B3Theorem B3: :

Let Let RR be a 2-torsion free and 3-torsion free semiprime ring be a 2-torsion free and 3-torsion free semiprime ring and and TT an endomorphism of an endomorphism of RR such that the mapping such that the mapping

xx [ [TT((xx)), x, x] is centralizing on ] is centralizing on RR. Then it is commuting . Then it is commuting on on RR; in particular, ([; in particular, ([TT((xx)), x, x])])22 = 0 for all = 0 for all xx R.R.

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5. Results (cont’d)5. Results (cont’d) Results -CResults -C

Proposition C1Proposition C1::

Let Let RR be a semiprime ring and be a semiprime ring and ff an epimorphism of an epimorphism of RR. . Then Then ff is centralizing if and only if it is is centralizing if and only if it is strong strong commutativitycommutativity-preserving-preserving. .

Proposition C2Proposition C2::

Let Let R R be a ring and be a ring and ff: : R R R R an antihomomorphism. an antihomomorphism. Then Then ff is commutativity-preserving. is commutativity-preserving.

Proposition C3Proposition C3::

Let Let RR be a 2-torsion free semiprime ring and be a 2-torsion free semiprime ring and ff a a centralizing antihomomorphism of centralizing antihomomorphism of RR onto itself. Then onto itself. Then f f is is strong commutativity-preserving.strong commutativity-preserving.

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5. Results (cont’d) -5. Results (cont’d) -CC

Bresar [10, Proposition 4.1] has proved the following result.Bresar [10, Proposition 4.1] has proved the following result.

TheoremTheorem

Let Let RR be a 2-torsion free semiprime ring and be a 2-torsion free semiprime ring and ff: : RR R R be be a centralizing antihomomorphism. Then a centralizing antihomomorphism. Then

i.i. S S = {= {xx R R: : ff((xx) ) = x= x} } ZZ((RR).).

ii.ii. If If RR is prime and is prime and ff does not map does not map R R into into ZZ((RR), then ), then S S == Z Z((RR). ).

RemarkRemark

We note that this Theorem can also be obtained as an We note that this Theorem can also be obtained as an application of Proposition C3 ifapplication of Proposition C3 if f f is onto. is onto.

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ReferencesReferences [1][1] H.E. Bell and W.S. Martindale, Centralizing mappings of H.E. Bell and W.S. Martindale, Centralizing mappings of

semiprime rings, Canad. Math. Bull. 30(1987), 92-101.semiprime rings, Canad. Math. Bull. 30(1987), 92-101. [2][2] M. Bresar, Centralizing mappings on Von Neumann M. Bresar, Centralizing mappings on Von Neumann

algebras, algebras, Proc. Amer. Math. Soc. III(1`991), 501-510.Proc. Amer. Math. Soc. III(1`991), 501-510. [3] Mr. Bresar and J. Vukman, On left derivations and related [3] Mr. Bresar and J. Vukman, On left derivations and related

mappings, Proc. Amer. Math. Soc. 100(1990), 7-16.mappings, Proc. Amer. Math. Soc. 100(1990), 7-16. [4][4] M. Choda, I. Kasahara and R. Nakamoto, Dependent M. Choda, I. Kasahara and R. Nakamoto, Dependent

elements elements of automorphisms of a C*-algebra, Proc. Japan of automorphisms of a C*-algebra, Proc. Japan Acad. Acad. 48(1972), 561-565.48(1972), 561-565.

[5][5] J.M. Cusack, Jordan derivations on rings, Proc. Amer. J.M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Math. Soc. 53(1975), 321-324.Soc. 53(1975), 321-324.

[6][6] I.N. Herstein, Jordan derivations in prime rings, Proc. I.N. Herstein, Jordan derivations in prime rings, Proc. Amer. Amer. Math. Soc. 8(1957), 1104-1110.Math. Soc. 8(1957), 1104-1110.

[7][7] R.R. Kallman, A generalization of free action, Duke Math. R.R. Kallman, A generalization of free action, Duke Math. J. J. 36(1969), 781-789.36(1969), 781-789.

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References (cont’d)References (cont’d) [8][8] L.A. Khan and A.B. Thaheem, On automorphisms of L.A. Khan and A.B. Thaheem, On automorphisms of

prime prime rings with involution, Demonst. Math. 30(1997), 307-311.rings with involution, Demonst. Math. 30(1997), 307-311. [9][9] Laradji and A.B. Thaheem, On dependent elements in Laradji and A.B. Thaheem, On dependent elements in

semiprime rings, Math. Japonica, 47(1998), 29-31.semiprime rings, Math. Japonica, 47(1998), 29-31. [10][10] J. Mayne, Centralizing automorphisms of prime rings, J. Mayne, Centralizing automorphisms of prime rings,

Canad. Math. Bull. 19(1976), 113-115.Canad. Math. Bull. 19(1976), 113-115. [11][11] J. Mayne, Centralizing mappings of prime rings, Canad. J. Mayne, Centralizing mappings of prime rings, Canad.

Math. Bull. 27(1984), 122-126.Math. Bull. 27(1984), 122-126. [12][12] J. D. P. Meldrum, Near-rings and their links with groups, J. D. P. Meldrum, Near-rings and their links with groups,

Research Notes in Maths. 134, Pitman, London, 1985.Research Notes in Maths. 134, Pitman, London, 1985. [13][13] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. E.C. Posner, Derivations in prime rings, Proc. Amer. Math.

Soc. 8(1957), 1093-1100.Soc. 8(1957), 1093-1100. [14][14] S. Sakai, C*-algebras and W*-algebras, Springer-Verlag, S. Sakai, C*-algebras and W*-algebras, Springer-Verlag,

Berlin, 1971.Berlin, 1971.

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References (cont’d)References (cont’d) [15][15] M.S. Samman, M.A. Chaudhry and A.B. Thaheem, A note M.S. Samman, M.A. Chaudhry and A.B. Thaheem, A note

on commutativity of automorphisms, Internat. J. Math. & on commutativity of automorphisms, Internat. J. Math. & Math. Sci. 21(1998), 201-204.Math. Sci. 21(1998), 201-204.

[16][16] A.M. Sinclair, Continuous derivations on Banach algebras, A.M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20(1969), 166-170.Proc. Amer. Math. Soc. 20(1969), 166-170.

[17][17] I.M. Singer and J. Wermer, Derivations on commutative I.M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129(1955), 260-264.normed algebras, Math. Ann. 129(1955), 260-264.

[18][18] S. Stratila, Modular Theory of Operator Algebras, Abcus S. Stratila, Modular Theory of Operator Algebras, Abcus Press, Kent, 1981.Press, Kent, 1981.

[19][19] M. Thomas, The image of a derivation is contained in the M. Thomas, The image of a derivation is contained in the radical, Ann. Math. 128(1988), 435-460.radical, Ann. Math. 128(1988), 435-460.

[20][20] J. Vukman, Derivations on semiprime rings, Bull. Austral. J. Vukman, Derivations on semiprime rings, Bull. Austral. Math. Soc. 53(1995), 353-359.Math. Soc. 53(1995), 353-359.

[21][21] J. Vukman, Commuting and centralizing mappings in J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109(1990), 47-52.prime rings, Proc. Amer. Math. Soc. 109(1990), 47-52.

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ThanksThanks

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Related to results -ARelated to results -A

A classical result in the theory of centralizing derivations is a A classical result in the theory of centralizing derivations is a theorem of Posner [36, Theorem 2] which states that theorem of Posner [36, Theorem 2] which states that noncommutative prime rings do not admit nonzero noncommutative prime rings do not admit nonzero centralizing derivations. centralizing derivations.

Vukman [48] has proved that if Vukman [48] has proved that if ff is a derivation on a is a derivation on a noncommutative prime ring noncommutative prime ring R R of characteristic not two such of characteristic not two such that the mapping that the mapping xx [ [dd((xx), ), xx] is commuting on ] is commuting on RR, then, then d d = 0. = 0. Alternatively, this result states that if Alternatively, this result states that if dd is a derivation which is a derivation which satisfies the identity satisfies the identity dd((xx))xx22 + + xx22dd((xx) - 2) - 2xdxd((xx))xx = 0 for all = 0 for all xx RR, then , then d d = 0. = 0. This , in fact, an analog of Posner's results for derivations This , in fact, an analog of Posner's results for derivations satisfying this identity.satisfying this identity.

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Related to results -BRelated to results -B

Certain situations have been identified where centralizing Certain situations have been identified where centralizing and commuting maps coincide. For instance, Bell and and commuting maps coincide. For instance, Bell and Martindale [3, Lemma 2] have shown that if Martindale [3, Lemma 2] have shown that if TT is an is an endomorphism of a semiprime ring endomorphism of a semiprime ring RR which is centralizing on which is centralizing on a nonzero left ideal a nonzero left ideal U U ofof R R, then , then TT is commuting on is commuting on U U..