5. Weakly primary - full

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.


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,


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


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.


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


.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.


(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 .


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


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.


Theorem: Let M be a multiplication entire partial semimodule over R and N be a subsemimodule of M. Then the


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.


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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