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WEAKLY PRIMARY SUBSEMIMODULES OF PARTIAL SEMIMODULES
M. SRINIVASA REDDY1, V. AMARENDRA BABU
2 & P. V. SRINIVASA RAO
3
1Assistant Professor, Department of S & H, D. V. R & Dr. H. S. MIC College of Technology, Krishna,
Andhra Pradesh, India
2Assistant Professor, Department of Mathematics, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India
3Associate Professor, Department of S & H, D. V. R & Dr. H. S. MIC College of Technology, Krishna,
Andhra Pradesh, India
ABSTRACT
The partial functions under disjoint-domain sums and functional composition is a so-ring, an algebraic structure
possessing a natural partial ordering, an infinitary partial addition and a binary multiplication, subject to a set of axioms. In
this paper we introduce the notion of weakly prime, weakly primary ideals in so-rings and weakly prime, weakly primary
subsemimodules in partial semimodules over partial semirings and we obtain the relation between them.
KEYWORDS: Weakly Prime Ideal, Weakly Primary Ideal, Weakly Prime Subsemimodule and Weakly Primary
Subsemimodule
INTRODUCTION
The study of ),( DDpfn (the set of all partial functions of a set D to itself), ),( DDMfn (the set of multi
functions of a set D to itself) and ),( DDMset (the set of all total functions of a set D to the set of all finite multi sets of
D) play an important role in the theory of computer science, and to abstract these structures Manes and Benson[1]
introduced the notion of sum ordered partial semirings (so-rings). In [6], we have obtained the ideal theory of so-rings. In
[7], [8] we have obtained some results on partial semimodules over partial semirings and we characterize prime and
semiprime subsemimodules with prime and semiprime partial ideals respectively. In this paper we obtain some
characteristics of weakly prime, weakly primary ideals of so-rings and characterize weakly prime & weakly primary
subsemimodules with weakly prime & weakly primary partial ideals of a partial semiring.
PRELIMINARIES
In this section we collect some important definitions and results for our use in this paper.
Definition: [5] A partial semiring is a quadruple )1,.,,( R , Where ),( R is a partial monoid with partial
addition ∑, )1,.,(R is a monoid with multiplicative operation and unit 1, and the additive and multiplicative structures
obey the following distributive laws. If ):( Iixi is defined in R, then for all y in R, ):( Iixy i and
):( Iiyxi are defined and i
i
i i
ii
i
i yxyxxyxy ).(][);(][
Throughout this paper we consider R as a commutative partial semiring.
International Journal of Mathematics and
Computer Applications Research (IJMCAR)
ISSN 2249-6955
Vol. 3, Issue 2, Jun 2013, 45-56
©TJPRC Pvt. Ltd.
46 M. Srinivasa Reddy, V. Amarendra Babu & P. V. Srinivasa Rao
Definition: [1] The sum ordering ≤ on a partial monoid ),( M is the binary relation ≤ such that x ≤ y if and
only if there exists a h in M such that y = x + h, for x, yM.
Definition: [1] A sum-ordered partial semiring (or so-ring for short), is a partial semiring in which the sum
ordering is a partial ordering.
Definition: [2] Let R be a so-ring. A subset N of R is said to be an ideal of R if the following are satisfied
(I1). if ):( Iixi is summable family in R and xiN for every i I then ∑xiN,
(I2). if x ≤ y and yN then xN, and
(I3). if xN and rR then xr, rxN.
A subset N of a partial semiring R satisfying (I1) and (I3) is called a partial ideal of R.
Definition: [2] A proper partial ideal P of a partial semiring R is said to be prime if and only if for any partial
ideals A, B of R, PAB implies PA or PB .
Spec(R) denotes the set of all prime partial ideals of a partial semiring R. For convenience we denote the set
}/)({ HIRspecH by )(IV and )(IV by I .
Theorem: [6] If I is a partial ideal of a commutative partial semiring R then IaRaI n /{ for some
positive integer n}.
Definition: [5] Let )1,,,( R be a partial semiring and ),( M be a partial monoid. Then M is said to be a left
partial semimodule over R if there exists a function xrxrMMR ),(:: which satisfies the following
axioms for x, ):( Iixi in M and r1, r2, ):( Jjr j in R
if iix exits then ),()( i
ii
ixrxr
if j
jr exits then xrj
j )( = ),( xr jj
,)()( 2121 xrrxrr
,1 xxR
MR x 00 .
Definition: [7] Let ),( M be a left partial semimodule over a partial semiring R. Then a nonempty subset N of
M is said to be a subsemimodule of M if and only if N is closed under and .
Definition: [7] Let N be a subsemimodule of a left partial semimodule M over R. Then
}/{}/):{():( NrMRrMmmNMN is called the associated partial ideal of N.
Weakly Primary Subsemimodules of Partial Semimodules 47
Definition: [8] Let M be a partial semimodule over R .Then M is said to be multiplication partial semimodule if
for any subsemimodule N of M there exists a partial ideal I of R such that N = I M.
Theorem: [8] A partial semimodule M over R is a multiplication partial semimodule if and only if there exists a
partial ideal I of R such that R m = I M for each mM.
Definition: [8] Let M be a multiplication partial semimodule over R and N, K be subsemimodules of M such that
N = I M and K = J M for some partial ideals I, J of R. Then the multiplication of N and K is defined as NK = (IM)(IJ) =
(IJ)M.
Definition: [8] Let M be a multiplication partial semimodule over R and m1, m2 in M such that R m1 = I1M and R
m2 = I2M for some partial ideals I1, I2 of R. Then the multiplication of m1 and m2 is defined as m1m2 = ( I1M) (I2M) =
(I1I2)M.
Definition: [8] Let M be a partial semimodule over R and N be a proper subsemimodule of M. Then N is said to
be prime subsemimodule of M if for any Rr and ,Mn Nnr implies ):( MNr or .Nn
Theorem: [8] Let M be a multiplication partial semimodule over R and N be a subsemimodule of M. Then N is
prime subsemimodule of M if and only if ):( MN is a prime partial ideal of R.
Theorem: [8] Let M be a multiplication partial semimodule over R and N be a subsemimodule of M. Then the
following conditions are equivalent:
N is prime subsemimodule,
for any subsemimodules U, V of M, NUV implies NU or NV ,
for any Mmm 21 , , Nmm 21 implies Nm 1 or Nm 2 .
WEAKLY PRIME & WEAKLY PRIMARY IDEALS
Definition: A proper ideal P of a so-ring R is said to be Weakly prime if for any ba, of R, Pab0 imply
Pa or Pb . Note that every prime ideal of a so-ring R is weakly prime. The following is an example of a so-ring R
in which weakly prime ideal is not prime.
Example: Consider the so-ring R = {0, u, v, x, y, 1} with ∑ defined on R by
.,
,,0,
otherwiseundefined
jsomeforjixifxx
ij
i
Table 1: And ‘ ∙ ‘ Defined by the Following
∙ 0 u v x y 1
0 0 0 0 0 0 0
u 0 u 0 0 0 u
v 0 0 v 0 0 v
x 0 0 0 0 0 x
y 0 0 0 0 0 y
1 0 u v x y 1
Then the ideal {0, x, y} is weakly prime. For },,,0{0,, yxvuRvu but },,0{ yxu and
},,0{ yxv . Hence {0, x, y} is not prime ideal of R.
48 M. Srinivasa Reddy, V. Amarendra Babu & P. V. Srinivasa Rao
Definition: A proper ideal P of a so-ring R is said to be weakly primary if for any a of R \ P, b of R ,
Pab0 imply Pb k for some positive integer k. Clearly every primary ideal of a so-ring R is weakly primary. The
following is an example of a so-ring R in which weakly primary ideal is not primary.
Example: Consider the so-ring R as in the example 2.2. Then the ideal },0{ u is weakly primary. For
},0{0},,0{ uxvux and nuvv n },0{ . Hence },0{ u is not primary ideal of R.
Clearly every weakly prime ideal of a so-ring R is weakly primary. The following is an example of a so-ring R in
which weakly primary ideal is not weakly prime.
Example: In the so-ring (the so-ring of all natural numbers with 0 and finite support addition), the ideal 4 is
weakly primary. Since 42240 and ,42 4 is not weakly prime ideal of R.
Definition: Let A and B be any subsets of a so-ring R. Then we define }/{):( ArBRrBA where
rbxRxrB /{ for some }Bb .
Note that }/{):( ArxRrxA and }0/{):0( rxRrx for any Rx .
Remark: If A and B are ideals of a so-ring R and Rx then ):( BA is an ideal of R and hence ):( xA and
):0( x are ideals of R.
Lemma: Let R be a so-ring. If an ideal I of R is the union of two ideals of R then it is equal to one of them.
Proposition: For a proper ideal A of a so-ring R, the following conditions are equivalent
A is weakly prime ideal of R,
for ),:0():(,\ xAXAARx
for AxAARx ):(,\ or ).:0():( xxA
Proof
(ii): Suppose A is weakly prime ideal of R and let ARx \ . Now for any ),:0( xAr Ar or
).:0( xr Arx or .0 Arx ):( xAr and hence ).:():0( xAxA Let ):( xAr .
Then Arx . If 0rx then ):0( xr . If 0rx then Ar (since A is weakly prime and Ax ).
):0( xAr . Hence ):0():( xAxA .
(iii): By lemma 2.8, we obtain (iii).
(i): Suppose for any AxAARx ):(,\ or ):0():( xxA . Let ARa \ and
AabRb 0 . Then ).:( aAb Ab or ):0( ab . If ):0( ab then 0ab , a
contradiction. Hence Ab . Therefore A is weakly prime ideal of R.
Proposition: For a proper ideal A of a so-ring R, the following conditions are equivalent
A is weakly primary ideal of R,
Weakly Primary Subsemimodules of Partial Semimodules 49
for ),:0():(,\ xAxAARx
for AxAARx ):(,\ or ):0():( xxA .
Proof
(ii): Suppose A is weakly primary ideal of R and let ARx \ . Then Ax n for every positive integer
n. Clearly ):():0( xAxA . Let ).:( xAr Then Arx . If 0rx then ):0( xr . If 0rx then
Ar (since Ax n and A is weakly primary). ):0( xAr . Hence ).:0():( xAxA
(iii): By lemma 2.8, we obtain (iii).
(i): Suppose AxA ):( or ):0():( xxA for any ARx \ . Let AxRr , such that
Arx0 . Then ):0( xr and ):( xAr . .):( AxA .Ar Hence A is weakly primary ideal
of R.
Lemma: Let A be an ideal of a so-ring R. If Aa and Aba then Ab .
Proof: Suppose Aa and Aba . Then Aa and Aba m )( for some positive integer m. Now
mmmmm bmabbmaaba 11 ...........)( . Since AmabmaacAa mmm 11 ......, . Since
Abcbcb mmm , and ,Ac we have Abm . Hence Ab .
Theorem: Let A be a weakly primary ideal of a so-ring R. If A is not primary then 02 A .
Proof: Suppose 02 A . Then iiaazAz '0 2 where Aaa ii ', . Let ., AxyRyx If 0xy
then Ax or Ay m for some positive integer m. Hence A is primary. So assume .0xy If AxA0 then
xadxAd 0 for some Aa . Ayaxxyxaxad )(0 . Since A is weakly primary,
Ax or Aya . By lemma 2.11, Ax or Ay .
Hence A is primary. So assume 0xA . Similarly we assume that 0yA . Now
Aayaxaaz iiii )')(('0 . Aayax ii )')((0 for some Ii . Aax i or
Aay i ' . Ax or Ay .
Hence A is primary. Hence the theorem.
Theorem: Let A be a weakly primary ideal of a so-ring R. If A is not primary then 0A .
Proof: Since AA 0,0 . By above theorem 02 A . 0 A . 0 A . Hence the
theorem.
Theorem: Let }/{ IiAi be a family of weakly primary ideals of a so-ring R that are not primary. Then
50 M. Srinivasa Reddy, V. Amarendra Babu & P. V. Srinivasa Rao
iIi
AA
is a weakly primary ideal of R.
Proof: Let AabRba 0,
and Ab . Then sAbIs and sAab0 .
AAa s 0 . Hence A is weakly primary.
The following theorem can be obtained similarly as above.
Theorem: Let A be a weakly prime ideal of a so-ring R. If A is not prime then 02 A .
Theorem: Let A be a weakly prime ideal of a so-ring R. If A is not prime then 0A .
Theorem: Let }/{ IiAi be a family weakly prime ideals of a so-ring R that are not prime. Then iIi
AA
is
a weakly prime ideal of R.
Lemma: If R is entire partial semiring then every weakly prime (weakly primary) partial ideal of R is prime
(primary).
Proof: Suppose R is entire and let I be a weakly prime (weakly primary ) partial ideal of R. Let
., IabRba If 0ab then either 0a or .0b Ia or Ib . If 0ab then Ia or Ib
( Ia or Ibn for some positive integer n). Hence I is prime (primary).
Corollary: If I is a weakly primary partial ideal of an entire partial semiring R then I is a weakly prime partial
ideal of R.
Proof: Suppose I is weakly primary. Then I is primary. I is prime. I is weakly prime partial ideal of
R.
WEAKLY PRIME & WEAKLY PRIMARY SUBSEMIMODULES
Throughout this section we consider R as a commutative partial semiring.
Definition: A subsemimodule N of a partial semimodule M over R is said to be weakly prime if
,,,0 MmRrNmr then either ):( MNr or Nm .
Definition: A subsemimodule N of a partial semimodule M over R is said to be weakly primary if
,,,0 MmRrNmr then either ):( MNr or Nm .
Clearly every prime subsemimodule of a partial semimodule M is primary and weakly prime and hence weakly
primary.
Following examples shows that the converse implications are not true.
Example: Consider the partial semiring R as in the example 2.2. Then },,0{ yx is a weakly prime
subsemimodule of RM .Since },,0{},,,0{0 yxuyxuv and },,0{},,,0{ yxyxv is not prime
subsemimodule of M.
Weakly Primary Subsemimodules of Partial Semimodules 51
Example: Consider the partial semiring R as in the example 2.2. Then },0{ u is a weakly primary
subsemimodule of RM . Since 0},,0{ xvux and },,0{ uvv n },0{ u is not primary subsemimodule of
M.
Example: In the partial semimodule (the so-ring of all natural numbers with 0 and finite support
addition), 4 is a weakly primary sumsemimodule. Since 42240 and 4,42 is not weakly prime
subsemimodule of .
Example: In the partial semimodule , 4 is a primary sumsemimodule. Since 4224 and
4,42 is not prime subsemimodule of .
Theorem: Let M be an entire partial semimodule over R and N be a weakly prime (weakly primary)
subsemimodule of M. Then its associated partial ideal ):( MN is weakly prime (weakly primary ) partial ideal of R.
Proof: Let ):(0 MNab and ):( MNa . Then NMab )(0 and NxaMx 0 .
0 xa . Since M is entire, .)()(0 NMabxab Nxab )(0 and Nxa . Since N is
weakly prime (weakly primary), ):( MNb ( ):( MNb ). Hence N is weakly prime (weakly primary) partial
ideal of R. The following is an example of a partial semimodule M over R in which the converse of above theorem is not
true in general.
Example: Let R be the partial semiring with finite support addition and usual multiplication. Then
M is an entire partial semimodule over R by the scalar multiplication ),()),(,(: xbxabax and
40K is a subsemimodule of M . Here }0{):( MK which is a weakly prime (weakly primary) partial ideal of
R. Since ):(2,)2,0(2)0,0( MKK ( ):(2 MKn for every positive integer n) and K)2,0( , K is not
weakly prime (weakly primary) subsemimodule of M.
The following example illustrates that if M is not entire then the above theorem will not be valid.
Example: Consider the partial semiring R and the partial semimodule ).,( 66 ZM Then {0} is a
weakly prime (weakly primary ) subsemimodule of M and 6):}0({ M . Now 63,6320 n and
62n for every positive integer n. Hence 6):}0({ M is not a weakly primary and hence not a weakly prime
partial ideal.
Theorem: If N is a weakly prime subtractive subsemimodule of a partial semimodule M, then either N is prime or
0):( NMN .
Proof: Suppose 0):( NMN . Let Rr and NmrMm . If 0mr then either
):( MNr or Nm . Hence assume 0mr . Suppose 0rN . Then
.0 nrNn Nnrnmr )(0 . ):( MNr or Nm . Hence assume 0rN .
Suppose .0):( mMN Then 0'):(' mrMNr and NMr ' . .')'(0 Nmrmrr
):(' MNrr or Nm . Since ):( MN is subtractive. ):( MNr or Nm . Hence assume
52 M. Srinivasa Reddy, V. Amarendra Babu & P. V. Srinivasa Rao
.0):( mMN Since ):(,0):( 1 MNrNMN and 0111 nrNn and NMr 1 .
Nnmrrnr )()(0 1111 . ):(1 MNrr or .1 Nnm
):( MNr or Nm .
Hence N is a prime subsemimodule of M.
Theorem: If N is proper subtractive subsemimodule of a partial semimodule M over R then the following
statements are equivalent:
whenever ,0 NID with I is a partial ideal of R and D is a subsemimodule of M, then either
):( MNI or ND ,
N is a weakly prime subsemimodule of M,
for ):0():():(,\ mMNmNNMm ,
for ):():(,\ MNmNNMm or ):0():( mmN .
Proof
(ii): Assume (i) and suppose Nmr 0 for some ., MmRr Take RrI and .RmD Then
NID0 . ):( MNI or ND . ):( MNr or Nm .
(i): Suppose N is weakly prime subsemimodule of M. If N is prime then we obtain (i) directly. Suppose N is
not prime. Then by theorem 3.10, .0):( NMN Now suppose .0 NDNID Then
NID0 and .NxDx Let Ir . If 0 xr then ).:( MNr So assume .0 xr
Suppose .0rD Then .0 NdrDd If Nd then ):( MNr . If Nd then
.)(0 Ndrxdr ):( MNr or Nxd . ):( MNr or Nx .
):( MNr . So assume that 0rD . Suppose 0Ix . Then .0 NxaIa
):( MNa . .)(0 Nxaxar ).:( MNar ):( MNr . So assume
that 0Ix . Since IbNID ,0 and NdbDd 11 0 .
.)(0 11 Ndbxdb ):( MNb or .1 Nxd If ):( MNb then Nxd 1 .
Now .0 1 Ndb .1 Nd ,Nx a contradiction. Hence ).:( MNb If Nxd 1
then ,0):()(0 11 NMNxdbdb a contradiction. Hence Nxd 1 . Since
):(,)()(0 11 MNbrNdbxdbr or Nxd 1 . ):( MNbr .
):( MNr for every Ir . ).:( MNI Hence (i).
(iii): Let .\ NMm Clearly ):():0():( mNmMN . Let ).:( mNa Then .Nma If
0ma then ):( MNa . If 0ma then ):0( ma . ).:0():( mMNa Hence
).:():0():( mNmMN
(iv): It is trivial.
Weakly Primary Subsemimodules of Partial Semimodules 53
(ii): Suppose ):():( MNmN or ):0():( mmN . Let NmrMmRr 0, and
.Nm ):( mNr and ).:0( mr ):( MNr . Hence N is weakly prime subsemimodule of M.
Theorem: If N be a proper subtractive subsemimodule of a partial semimodule M then the following statements
are equivalent:
N is a weakly primary subsemimodule of M,
for ).:0():():(,\ mMNmNNMm
Proof
(ii): Let NMm \ . Then ).:():0():( mNmMN Let .):( mNa Then Nmak
for some positive integer k. If 0ma k then ):()( MNa lk for some positive integers lk, .
):( MNa . If 0ma k. Let s be the smallest positive integer such that 0ma s
. If s = 1 then
):0( ma . Otherwise 0ma . ):( MNa . Hence ).:0():():( mMNmN
(i): Suppose ):0():():( mMNmN for NMm \ . Let
.0\, NmrNMmRr ):():( mNmNr and ).:0( mr ):( MNr .
Hence N is weakly primary subsemimodule of M.
Theorem: If N is weakly primary subtractive subsemimodule of a partial semimodule M then either N is primary
or 0):( NMN .
Proof
Suppose 0):( NMN . Let ., NmrMmRr If 0mr then we obtain the conclusion. So
assume that 0mr . Suppose .0rN Then 0 nrNn . Nnrnmr )(0 .
):( MNr or Nnm . ):( MNr or .Nm So we can assume that 0rN . Suppose
.0):( mMN Then .0):( 11 NmrMNr Nmrmrr 11)(0 .
):(1 MNrr or Nm . ):( MNr or Nm (lemma 2.11 over partial semirings). So we can
assume that 0):( mMN . Since 0):( NMN , ):(2 MNr and NnrNn 121 0 .
.)()(0 1212 Nnrnmrr ):(2 MNrr or Nnm 1 . ):( MNr or
Nm . Hence N is primary subsemimodule of M.
Theorem: Let M be a multiplication entire partial semimodule over a partial semiring R and N be a
subsemimodule of M. Then N is weakly prime (weakly primary) subsemimodule of M if and only if its associated partial
ideal ):( MN is weakly prime (weakly primary) partial ideal of R .
54 M. Srinivasa Reddy, V. Amarendra Babu & P. V. Srinivasa Rao
Proof: By theorem 3.10, we get the necessary part. For sufficient part, suppose ):( MN is weakly prime
(weakly primary) partial ideal of R. Let NxaMxRa 0, and Nx .
Since M is multiplication partial semimodule, a partial ideal I of R such that IMRx .
Now NxaRRxaIMaMaI )()()()( and .NIMRx ):( MNaI and
):( MNI . Suppose 0aI . Then 0)( MaI . .0)*( xaR ,0* xa a contradiction. Hence
0aI . Thus we have ):(),:(0 MNIMNaI and ):( MN is weakly prime (weakly primary ) partial
ideal of R. ):( MNa ( ):( MNa ). Hence N is weakly prime (weakly primary) subsemimodule of M.
Theorem: Let M be a multiplication entire partial semimodule over R and N be a subsemimodule of M. Then the
following conditions are equivalent:
N is weakly prime subsemimodule of M,
for any subsemimodules U, V of M, NUV 0 implies NU or NV ,
for any NmmMmm 2121 0,, implies Nm 1 or Nm 2 .
Proof
(ii): Suppose N is a weakly prime subsemimodule of M and let U, V be subsemimodules of M such that
NUV 0 . Since M is multiplication partial semimodule, partial ideals I, J of R IMU and
JMV . Now NMIJUV )(0 . ).:(0 MNIJ Since ):( MN is weakly prime (by
theorem 3.10), ):( MNI or ):( MNJ . NIMU or NJMV .
(iii): Suppose for any subsemimodules U, V of M, NUV 0 implies NU or NV . Let
NmmMmm 2121 0, . Since M is multiplication partial semimodule, partial ideals I, J of R
IMRm 1 and JMRm 2 . Now NMIJRmRmmm )())((0 2121 . NRm 1 or
NRm 2 . Nm 1 or Nm 2 .
(i): Suppose for any NmmMmm 2121 0,, implies Nm 1 or Nm 2 . We prove that
):( MN is weakly prime partial ideal of R. Let I, J be a partial ideals of R ):(0 MNIJ and
):(),:( MNJMNI . NIMNMIJ ,)( and NJM .
NIMmiMmmJjIi \,,, 121 and NJMmj \2 .
NMIJJMIMmjmi )())(())(( 21 . Since NJMmjNIMmi \,\ 21 . We have
01 mi and 02 mj . Since M is entire, 0))(( 21 mjmi . Nmi 1 or Nmj 2 , a
contradiction . Hence ):( MN is weakly prime partial ideal of R. N is weakly prime subsemimodule of M.
Weakly Primary Subsemimodules of Partial Semimodules 55
Theorem: Let M be a multiplication entire partial semimodule over R and N be a subsemimodule of M. Then the
following conditions are equivalent:
N is weakly primary subsemimodule of M,
for any subsemimodules U, V of M, NUV 0 implies NU or ‘ Nvn for some positive integer n,
Vv ’,
for any NmmMmm 2121 0,, implies Nm 1 or Nmn2 for some positive integer n.
Proof
(ii): Suppose N is a weakly primary subsemimodule of M and let U, V be subsemimodules of M
.0 NUV Since M is multiplication partial semimodule, partial ideals I, J of R IMU and
JMV . Now NMIJUV )(0 . ):(0 MNIJ . Since ):( MN is weakly primary (by
theorem 3.10), ):( MNI or ‘ ):( MNj n for some positive integer n, Jj ’. NIMU or
‘ NMjjMv nnn )( for some positive integer n, Vv ’.
(iii): Suppose for any subsemimodules U, V of M, NUV 0 implies NU or ‘ Nvn for some
positive integer n, Vv ’. Let .0, 2121 NmmMmm
Since M is multiplication partial
semimodule, partial ideals I, J of R IMRm 1 and JMRm 2 . Now
NMIJRmRmmm )())((0 2121 . NRm 1 or Nmr n )( 2 for some positive integer n,
JMRmmr 22 . Nm 1 or Nmn2 for some positive integer n.
(i): Suppose for any NmmMmm 2121 0,, implies Nm 1 or Nmn2 for some positive
integer n. We prove that ):( MN is weakly primary partial ideal of R. Let I, J be partial ideals of R
):(0 MNIJ and ):(),:( MNjMNI n for some Jj and for every positive integer n.
NIMNMIJ ,)( and NMj n . NIMmiMmmIi \0,, 121 and
NMjmj nn \0 2 . .)())(())((0 21 NMIJJMIMmjmi Nmi 1 or
Nmj n )( 2 for some positive integer n, a contradiction . Hence ):( MN is weakly primary partial ideal of R.
N is weakly primary subsemimodule of M.
Corollary: Let N be a weakly primary subsemimodule of an entire partial semimodule M over R. Then
):( MN is a weakly prime partial ideal of R containing ):0( M .
Corollary: If N is weakly primary subsemimodule of an entire multiplication partial semimodule M then N is
weakly prime subsemimodule of M.
56 M. Srinivasa Reddy, V. Amarendra Babu & P. V. Srinivasa Rao
CONCLUSIONS
In this paper we defined weakly prime & weakly primary ideals of so-rings and observed equivalent conditions
for an ideal to be weakly prime & weakly primary. Also we introduced the notion of weakly prime & weakly primary
subsemimodules for a partial semimodule over a partial semirings and obtained the equivalent conditions. In multiplication
partial semimodule we characterized weakly prime & weakly primary subsemimodules with weakly prime & weakly
primary partial ideals of a partial semiring.
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