Vol. 19, No. 1 (2013), pp. 49–59.
ON SEMIPRIME SUBSEMIMODULES
AND RELATED RESULTS
Hamid Agha Tavallaee
1and Masoud Zolfaghari
21Department of Mathematics, Tehran-North Branch,
Islamic Azad University, Tehran, Iran [email protected]
2Department of Mathematics, Semnan University, Semnan, Iran
Abstract. In this paper we introduce the notions of semiprime subsemimodules
and semiprimek-subsemimodules and present some characterizations about them. Special attention has been paid, when semimodules are multiplication, to find ex-tra properties of these semimodules. Moreover we prove a result for semiprime subsemimodules of quotient semimodules.
Key words: Semiprime subsemimodule, Strong semiprime k-subsemimodule, S-semiring, S-semimodule.
Abstrak.Pada paper ini kami memperkenalkan ide-ide subsemimodul semiprima,
subsemimodul-k semiprima, dan menyajikan beberapa sifat mereka. Perhatian khusus diberikan, saat semimodul adalah perkalian, untuk menemukan sifat-sifat tambahan dari semimodul ini. Lebih jauh, kami membuktikan sebuah hasil untuk subsemimodul semiprima dari semimodul hasil bagi.
Kata kunci: Subsemimodul semiprima, semiprima kuat, semimodul-k, semiring-S, semimodul-S.
1. Introduction
In the recent years a good deal of researches have been done concerning semir-ings and semimodules (for example see [1]-[6]). Particularly, there are numerous applications of semirings and semimodules in various branches of mathematics and computer sciences (for example see [6]). Also, in the last decade the notion of prime subsemimodules has been studied by many authors (for example see [3], [6] and [8]). Having the vast heritage of ring theory available, a number of authors have tried
2000 Mathematics Subject Classification: 16P20.
to extend and generalize many classical notions and definitions. For example, from the definition of a prime submodule, they have reached to prime subsemimodule etc. In this paper we define semiprime subsemimodules and strong semiprime k -subsemimodules of semimodules and prove some of their properties. In section 2, we recall some of the known notions and definitions. In sections 3, we give some basic results about semiprime subsemimodules of semimodules and quotient semi-modules. Section 4 is devoted to the study of strong semiprimek-subsemimodules,
S-semirings andS-semimodules.
2. Preliminaries
We begin this section with some necessary definitions.
Definition 2.1. (a) A non-empty set R together with two binary operations (called addition and multiplication and denoted by +, · respectively) is called a semiring provided that:
(i) (R,+) is a commutative semigroup (ii) (R,·) is a semigroup
(iii) There exists 0∈R such thatr+ 0 =randr·0 = 0·r= 0 for allr∈R
(iv) Multiplication distributes over addition both from the left and right.
IfR contains the multiplicative identity 1, thenR is called a semiring with identity. A semiringRis commutative if (R,·) is a commutative semigroup. In this paper all semirings are commutative with identity.
(b)A non-empty subset I of a semiringR is called anideal ofR ifa, b∈I
andr∈Rimplies thata+b∈I andra, ar∈I.
(c) An ideal I of a semiring R is called a subtractive ideal (or k-ideal) if
a, a+b∈Iimplies thatb∈I. For example{0}is a k-ideal ofR.
(d)An ideal I of a semiring R is called apartitioning ideal (or Q-ideal) if there exists a subset QofRsuch that R=∪{q+I|q∈Q} and ifq1, q2∈Qthen (q1+I)∩(q2+I)̸=∅if and only if q1=q2.
Definition 2.2. (a)Asemimodule M over a semiringR(or anR-semimodule) is a commutative monoid (M,+) with additive identity 0M, together with a function R×M −→M, defined by (r, m)7→rm(called scalar multiplication) such that:
(i) r(m+m′) =rm+rm′ (ii) (r+r′)m=rm+r′m (iii) (r·r′)m=r(r′m) (iv) 1Rm=m
(v) 0Rm=r0M = 0M for allr, r′
∈Randm, m′
∈M. Clearly, every semiring is a semimodule over itself.
(b)A subsetN of the R-semimodule M is called a subsemimodule of M if
(c)LetM be a semimodule over a semiringR. Asubtractive subsemimodule (or k-subsemimodule) N is a subsemimodule ofM such that if a, a+b∈N, then
b∈N. For example {0M}is a k-subsemimodule ofM.
(d)A subsemimodule N of a semimodule M over a semiring R is called a partitioning subsemimodule (orQ(M)-subsemimodule) if there exists a non-empty subsetQ(M) ofM such that:
(i) RQ(M)⊆Q(M), whereRQ(M) ={rq|r∈R, q∈Q(M)}
(ii) M =∪{q+N|q∈Q(M)}
(iii) Ifq1, q2∈Q(M) then (q1+N)∩(q2+N)̸=∅if and only if q1=q2.
Since every semiring is a semimodule over itself, every partitioning ideal of a semiringR is a partitioning subsemimodule of theR-semimoduleR.
Example 2.3. Let R = {0,1,2,· · ·, n} and define x+y = max{x, y} and
xy =min{x, y} for each x, y ∈ R. By [1], Example 5, R together with the two defined operations forms a semiring. LetM denote the set of all non- negative inte-gers. Definea+b=max{a, b}for eacha, b∈M. It is easy to show that (M,+) is a commutative monoid with identity 0. Define a function fromR×M in toM, sending (r, m) tomin{r, m}(r∈R, m∈M). It is easy to see thatM is anR-semimodule. Now we show thatN =Ris anR-subsemimodule ofM. Leta, b∈N, r∈R. Since
a, b≤n, soa+b=max{a, b} ≤n. Hencea+b∈N. Also we havera=min{r, a}. Ifa≤r thenra=a∈N and if a > r, ra=r∈N. It is clear from the definition of addition inM that 0 +N =N andk+N ={k}for eachn < k(n∈N). Thus
N is aQ(M)-subsemimodule ofM whenQ(M) ={0}∪{k∈M|n < k}.
Proposition 2.4. Let R be a semiring, M an R-semimodule and N a Q(M) -subsemimodule ofM. Then N is ak-subsemimodule of M.
Proof. Letx, x+y∈N. SinceM =∪{q+N|q∈Q(M)} we can writey=q+z for some q∈Q(M) andz∈N. Thereforex+y =x+q+z=q+x+z∈q+N. Alsox+y∈0M +N. Hence (q+N)∩(0M +N)̸=∅ and soq= 0M. Therefore y∈N, as required.✷
Definition 2.5. LetNbe a subsemimodule of anR-semimoduleM. Then (N :M) is defined as (N:M) ={r∈R|rM ⊆N}. Clearly (N :M) is an ideal ofR. The annihilator of M is defined as (0 :M) and is denoted byann(M). Ifann(M) = 0 thenM is called faithful.
Definition 2.6. LetM andM′
be semimodules over the semiringR. A functionf :
M −→M′
ofR-semimodules is a one-to-one and onto homomorphism ofR-semimodules. The
R-semimodulesM and M′ are called isomorphic and are denoted byM ∼ =M′ if there exists an isomorphism fromM toM′.
Remark 2.7. LetM,M′are semimodules over the semiringRandf :M
−→M′ be a homomorphism of R-semimodules. Then kerf is ak-subsemimodule of M. Because ifx, x+y ∈kerf, we can writef(x+y) =f(x) +f(y). Hence 0 = 0 +f(y) and so y∈kerf, as required.
3. Semiprime Subsemimodules
LetM be a semimodule over a semiringR. A proper subsemimoduleN of
M is called prime if for eachr∈R andm∈M,rm∈N implies thatr∈(N :M) orm∈N.
IfN is a prime subsemimodule ofM, then (N:M) is a prime ideal ofR(see [3], Lemma 4). As it is known a good deal of research has been done for semiprime submodule of a module during the last two decades. In this section we define semiprime subsemimodules and try to find some of their essential properties.
Definition 3.1. A proper subsemimodule N of an R-semimodule M is called semiprime if for eachr∈R,m∈M and positive integer t,rtm
∈N implies that
rm∈N.
Since the semiring R is anR-semimodule by itself, according to our defini-tion, a proper ideal I of R is a semiprime ideal, if whenever atb
∈ I for every
a, b ∈ R and positive integer t, then ab ∈ I. If the semiprime subsemimo-dule N of M is a k-subsemimodule (Q(M)-subsemimodule), then N is called a semiprimek-subsemimodule (semiprime Q(M)-subsemimodule). In a similar way if the semiprime ideal I of R is a k-ideal (Q-ideal), then I is called a semiprime
k-ideal (semiprimeQ-ideal).
Proposition 3.2. Let R be a semiring, M an R-semimodule and N a proper subsemimodule ofM. ThenN is semiprime if and only ifr2m∈N(r∈R, m∈M) implies that rm∈N.
Proof. LetN be a semiprime subsemimodule ofM andr2m∈N, wherer∈R and m∈M. Then by definitionrm∈N. Conversely, letrtm
∈N, wherer∈R,
m∈M andt∈Z+. Thenr2(rt−2m)∈N and by hypothesisr(rt−2m) =rt−1m∈ N. we writert−1m=r2(rt−3m)∈N and sor(rt−3m) =rt−2m∈N. In this way we finally obtainr2m∈N which implies thatrm∈N.✷
I= (p1· · ·pk) = (p1)∩· · ·∩(pk), wherep1,· · ·pk are distinct prime numbers.
Proposition 3.4. Let R be a semiring andM anR-semimodule. IfN is a prime subsemimodule ofM, then N is a semiprime subsemimodule ofM.
Proof. Letrtm
∈ N, where r ∈ R, m ∈ M and t ∈ Z+. Since N is prime so
m ∈N or rt
∈(N :M). If m∈N thenrm ∈N. Now let rt
∈(N : M). Since (N : M) is a prime ideal of R, so r ∈(N : M). Hence rm ∈N. In any case we have rm∈N and soN is a semiprime subsemimodule ofM.✷
Proposition 3.5. Let R be a semiring and M an R-semimodule. If N is a semiprime subsemimodule ofM, then(N :M) is a semiprime ideal ofR.
Proof. Since N is a proper subsemimodule of M, so (N : M) ̸= R. Let
atb
∈ (N :M) in which a, b∈R and t∈ Z+. Then atbM
⊆N and so for every
m∈M,atbm
∈N. SinceN is a semiprime subsemimodule ofM, soabm∈N and henceab∈(N :M), as required.✷
Definition 3.6. A semiring R is said to be an integral semidomain if for every
a, b∈R,ab= 0 implies that a= 0 orb= 0.
Example 3.7. It is clear that Z+ is an integral semidomain.
In the next example we show that the converse of Proposition 3.5 is not true in general.
Example 3.8. Let R = Z+. Then M = Z+⊕Z+ is an R-semimodule, N =
{r(9,0)|r ∈ R} is a subsemimodule of M and (N : M) = 0. Since R is an inte-gral semidomain, (N : M) = 0 is a prime and hence semiprime ideal of R. But
N is not a semiprime subsemimodule of M, because 32(2,0) = (18,0)
∈ N and
3(2,0) = (6,0)∈/N.
The next theorem gives a characterization of semiprime subsemimodules. This is specially useful when we work with multiplication semimodules.
Theorem 3.9. Let R be a semiring, M an R-semimodule and N a proper sub-semimodule of M. Then N is semiprime if and only if for every ideal I of R, subsemimodule K ofM and positive integert,ItK
⊆N implies that IK ⊆N.
Proof. LetIbe an ideal ofRandKa subsemimodule of M such thatItK
⊆N, wheret∈Z+. Consider the setS ={ax|a∈I, x∈K}. ThenRa is an ideal ofR,
Rxis a subsemimodule ofM and we have (Ra)t(Rx)
⊆ItK
⊆N. Henceatx
∈N
must haveIK⊆N. Conversely, leta∈R, m∈M andatm
∈N in which t∈Z+. TakeI=RaandK=Rm. Then we haveItK
⊆N. HenceIK⊆N which implies that am∈N. The proof is now completed. ✷
Remark 3.10. Note that since the semiringR is anR-semimodule by itself, then by Theorem 3.9, I is a semiprime ideal of R, if and only if for every idealsJ and
K ofRand positive integert,JtK
⊆I implies thatJK ⊆I.
Multiplication modules play an important rule in module theory. We recall
Definition 3.11. Let R be a semiring. An R-semimodule M is called multi-plication semimodule provided that for every subsemimoduleN ofM there exists an ideal IofR such thatN =IM.
Now we show that ifM is a multiplication semimodule then the converse of Proposition 3.5, is true.
Proposition 3.12. Let M be a multiplication semimodule over a semiringR and N a proper subsemimodule ofM. ThenN is a semiprime subsemimodule of M if and only if(N :M)is a semiprime ideal of R.
Proof. Let (N :M) be a semiprime ideal ofR. Assume thatItK
⊆N in whichI
is an ideal ofR,Ka subsemimodule ofM andt∈Z+. SinceM is a multiplication
R-semimodule, we can writeK=JM for some idealJ ofR. ThereforeItJM
⊆N
and soItJ
⊆(N :M). Since (N :M) is semiprime,IJ ⊆(N :M). From this we have IJM ⊆N and so IK ⊆N. Hence N is a semiprime subsemimodule of M. The converse is true by Proposition 3.5.✷
Proposition 3.13. LetM be a semimodule over a semiringRandN a semiprime k-subsemimodule of M. Then(N :M) is a semiprimek-ideal ofR.
Proof. By Proposition 3.5, (N :M) is a semiprime ideal ofR. Now letx, x+y∈ (N :M). Then for eachm∈M we have xm,(x+y)m=xm+ym∈N. Hence, sinceN isk-subsemimodule,ym∈N. Thereforey ∈(N:M) and the assertion is proved.✷
The next lemma gives conditions for a family of semiprime subsemimodules to be semiprime.
Lemma 3.14. Let {Nγ}γ∈Γ be a non-empty family of semiprime subsemimod-ules of a semimodule M over a semiring R. Then N =∩γ∈ΓNγ is a semiprime subsemimodule ofM. If, in addition,{Nγ}γ∈Γ is totally ordered by inclusion, then T =∪γ∈ΓNγ is a semiprime subsemimodule ofM provided that T ̸=M.
Proof. By [5], Lemma 1,N is a subsemimodule ofM. Letrtm
∈N, wherer∈R,
m∈M andt ∈Z+. Then for every γ∈Γ, rtm
∈Nγ. SinceNγ is semiprime, we haverm∈Nγ and this is true for everyγ∈Γ. Thereforerm∈
∩
fact that{Nγ}γ∈Γis totally ordered by inclusion makes it clear thatTis a subsemi-module ofM and we can simply show thatTis a semiprime subsemimodule ofM.✷
Definition 3.15. Let R be a semiring and I an ideal of R. The radical of I, denoted by √I, is the set of allx∈ R such that there exists a positive integern
(depending on x) withxn
∈I. The idealIofRis called a radical ideal ifI=√I.
Lemma 3.16. Let R be a semiring and P a semiprime ideal ofR. Then P is a radical ideal .
Proof. ClearlyP ⊆√P. Leta ∈√P be an arbitrary element. Then for some
n∈Z+,an =an.1
∈P. SinceP is semiprime, soa=a.1∈P, as required.✷
The notion of primary submodule is fundamental when we work with mod-ules. Similarly we define
Definition 3.17. Let M be a semimodule over a semiring R. A proper subse-mimoduleN ofM is calledprimary if for eachr∈Randm∈M,rm∈N implies that m ∈ N or rn
∈ (N : M) for some positive integer n. Clearly every prime subsemimodule is primary.
Since the semiring R is anR-semimodule by itself, according to our defini-tion, a proper idealIofR is aprimary ideal, if wheneverab∈Ifor each a, b∈R, then a ∈I or bn
∈ I for some positive integern. If the primary subsemimodule
N ofM is ak-subsemimodule (Q(M)-subsemimodule) thenN is called primaryk -subsemimodule (primary Q(M)-subsemimodule). Prime k-subsemimodule (prime
Q(M)-subsemimodule) is defined in a similar fashion. Also we can define prime and primaryk-ideal (Q-ideal) in the same way.
Definition 3.18. Let R be a semiring and M an R-semimodule. M is called a cancellative semimodule if wheneverrm=smfor elementsm∈M andr, s∈R, thenr=s.
A semiringR is called acancellative semiring if it is a cancellative semimo-dule over itself.
Proposition 3.19. ([3], Proposition 1) Let R be a semiring, M a cancellative R-semimodule and N a proper Q(M)-subsemimodule of M. Then (N : M) is a Q-ideal ofR.
Theorem 3.20. Let Rbe a semiring andM a cancellativeR-semimodule. IfN is a semiprimeQ(M)-subsemimodule of M, then(N :M)is a semiprime Q-ideal of R.
Proof. This is clear by using Proposition 3.4 and Proposition 3.17.✷
a proper Q(M)-subsemimodule of M. Then N is a prime Q(M)-subsemimodule if and only if N is primary Q(M)-subsemimodule and (N : M) is a semiprime Q-ideal ofR.
Proof. LetN be a primeQ(M)-subsemimodule ofM. ThenN is primaryQ(M )-subsemimodule and by Propositions 3.3, 3.4 and 3.17, (N : M) is a semiprime
Q-ideal of R. Conversely, let rm ∈ N where r ∈ R and m ∈ M. Assume that Definition 2.2(d), because if there exists alsoq′
∈Q(M) suchx+N ⊆q′+N then subsemimodule of M. By Lemma 3.22, we can define a binary operation ⊕ on
{q+N|q∈Q(M)}as follows:
Theorem 3.24. ([2], Theorem 2.4)Let R be a semiring,M an R-semimodule, N a subsemimodule ofM andQ1(M)andQ2(M)non-empty subsets ofM such that N is both a Q1(M)-subsemimodule andQ2(M)-subsemimodule. Then({q+N|q∈ Q1(M)},⊕,⊙)∼= ({q+N|q∈Q2(M)},⊕,⊙).
Remark 3.25. (Quotient semimodule) If R is a semiring, M an R-semimodule andN a subsemimodule ofM, then it is possible thatN can be considered to be a
Q(M)-subsemimodule with respect to many different subsetsQ(M) ofM. However, Theorem 3.24, implies that the structure ({q+N|q∈Q(M)},⊕,⊙) is essentially independent of the choice ofQ(M). Thus, if N is a Q(M)-subsemimodule ofM, the semimodule ({q+N|q∈Q(M)},⊕,⊙) is called a quotient semimodule ofM
there exists a unique elementq0 ∈Q(M) such thatq0+N =N. Thus q0+N is the zero element ofM/N.
To prove our next result we need the following three theorems.
Theorem 3.26. ([3], Theorem 1) Let R be a semiring, M an R-semimodule, N aQ(M)-subsemimodule of M andL ak-subsemimodule of M withN ⊆L. Then L/N ={q+N|q∈L∩Q(M)} is ak-subsemimodule ofM/N.
Theorem 3.27. ([3], Theorem 2) Let R be a semiring, M an R-semimodule, N aQ(M)-subsemimodule ofM andL ak-subsemimodule ofM/N. ThenLis in the formT /N, whereT is ak-subsemimodule of M which containsN.
Theorem 3.28. ([3], Theorem 4) Let R be a semiring, M an R-semimodule, N aQ(M)-subsemimodule of M and let T, Lbe k-subsemimodules of M contain-ingN. ThenT /N =L/N if and only ifT =L.
Theorem 3.29. Let N be a Q(M)-subsemimodule of a semimodule M over a semiringRandT be ak-subsemimodule ofM withN⊆T. ThenT is a semiprime R-subsemimodule of M if and only if T /N is a semiprime R-subsemimodule of M/N.
Proof. LetT be a semiprime R-subsemimodule ofM. By Definition 3.1,T is a proper subsemimodule ofM and so,T /N ̸=M/N, by Theorem 3.28. To show that
T /N is a semiprimeR-subsemimodule ofM/N, letrt(q
1+N) =rtq1+N ∈T /N whereq1∈Q(M),r∈Randt∈Z+. It follows from Theorem 3.26, thatrtq1∈T and sinceT is semiprime, we have rq1 ∈T. Hencer(q1+N)∈T /N, as required. Conversely, let T /N be a semiprime R-subsemimodule of M/N. Hence T /N is a proper R-subsemimodule ofM/N and soT ̸=M, by Theorem 3.28. Letrla
∈T
wherer∈R,a∈M andl∈Z+. Sincea∈M andN is aQ(M)-subsemimodule of
M, there are elementsq∈Q(M) andn∈N such thata=q+n, by Lemma 3.22. So rla=rlq+rln
∈ T. SinceT is a k-subsemimodule of M and rln
∈ N ⊆T, so rlq
∈T. Therefore,rl(q+N) =rlq+N
∈ T /N. Then T /N semiprime gives
r(q+N)∈T /N and sorq+N ∈T /N. Hencerq∈T and sora∈T. The proof is now completed.✷
4. Strong Semiprime k-Subsemimodules
In this section we investigate strong semiprimek-subsemimodules will lead us to some interesting results. First we recall some definitions.
thenIis called astrong semiprime k-ideal. If the strongk-idealIofRis a radical ideal, then we callI a strongk-radical ideal.
(b)A subsemimoduleN of an R-semimoduleM is said to be astrong sub-semimodule if for each a ∈ N there exists b ∈ N such that a+b = 0. Clearly, every submodule of a module over a ring R is a strong subsemimodule. If a strong subsemimodule N of M is a k-subsemimodule, then we call N a strong k-subsemimodule. If a strong k-subsemimodule N of M is semiprime, then N is called astrong semiprime k-subsemimodule.
Proposition 4.2. ([5], Proposition 1) LetM be a semimodule over a semiringR. Then the following statements hold:
(i) If N is a strong subsemimodule ofM, thenN is ak-subsemimodule. (ii) If I is a strong ideal of R, thenIM is a strongk-subsemimodule.
Example 4.3. The monoidM = (Z6,+6) is a semimodule over (Z+,+,·). We can show that N ={0,2,4} is a strongQ(M)-subsemimodule of M (and sok-strong subsemimodule of M, by Proposition 4.2(i)), whereQ(M) ={0,1}.
Lemma 4.4. ([4], Proposition 2)Let M be a finitely generated semimodule over a semiringR andIbe a strong k-radical ideal ofR. Then(IM :M) =I if and only if ann(M)⊆I.
Proposition 4.5. Let M be a finitely generated semimodule over a semiring R, P a strongk-radical ideal ofR containing ann(M)and I an ideal of R such that IM ⊆P M. ThenI⊆P.
Proof. By Lemma 4.4, we have (P M :M) = P. Now letr∈I be an arbitrary element. ThenrM⊆IM ⊆P M which impliesr∈(P M :M) =P, as required.✷
Theorem 4.6. Let M be a finitely generated multiplication semimodule over a semiringRandP a strong semiprimek-ideal ofR containingann(M). ThenP M is a strong semiprime k-subsemimodule of M.
Proof. By Lemma 3.16,Pis a radical ideal ofRand so (P M :M) =P, by Lemma 4.4. On the other hand, by Proposition 4.2(ii),P M is a strongk-subsemimodule of
M. Now it is enough to show that P M is a semiprime subsemimodule ofM. Let
I be an ideal ofR, K a subsemimodule ofM and t ∈Z+ such that ItK
⊆P M. Since M is a multiplication R-semimodule, there exists an idealJ ofR such that
K = JM. Hence ItJM
⊆ P M and by Proposition 4.5, ItJ
Definition 4.7. (a)A semiringR is called anS-semiring if every proper ideal in
R is a product of strong semiprimek-ideals.
(b)A semimoduleM over a semiringR is called an S-semimodule if every proper subsemimodule N of M either is strong semiprime k-subsemimodule or has a factorization in the form N = P1· · ·PnN⋆ in which P1,· · ·, Pn are strong semiprimek-ideals ofRandN⋆ is a strong semiprimek-subsemimodule ofM.
Example 4.8. A Dedekind domain is an integral domainRin which every proper ideal is the product of a finite number of prime ideals. LetM be a multiplication module over a Dedekind domainR. ThenM is clearly anS-semimodule.
Theorem 4.9. Let M be a faithful finitely generated multiplication semimodule over an S-semiringR. ThenM is anS-semimodule.
Proof. LetN be a proper subsemimodule ofM. Since M is a multiplication se-mimodule, we can writeN =IM for some idealI ofR. SinceR is anS-semiring, so I = P1· · ·Pn, where Pi(1 ≤ i ≤ n) is a strong semiprime k-ideal of R and so N =P1· · ·PnM. By Theorem 4.6, PiM(1 ≤i ≤n) is a strong semiprime k -subsemimodule ofM. Therefore N either is a strong semiprime k-subsemimodule ofM or has a factorization in the formN =P1· · ·Pn−1N⋆, whereN⋆=PnM. ✷
References
[1] Allen, P.J., ”A Fundamental Theorem of Homomorphisms for Semirings”,Proc. Amer. Math. Soc., 21(1969), 412 - 416.
[2] Chaudhari, J.N. and Bonde, D., ”On Partitioning and Subtractive Subsemimodules of Semi-modules over Semirings”,Kyungpook Math. J.,50(2010), 329 - 336.
[3] Ebrahimi Atani, R. and Ebrahimi Atani, S., ”On Subsemimodules of Semimodules”,Buletinul Academiei De Stiinte,2(63) (2010), 20 - 30.
[4] Ebrahimi Atani, R. and Ebrahimi Atani, S., ”Spectra of Semimodules”,Buletinul Academiei De Stiinte,3(67) (2011), 15 - 28.
[5] Ebrahimi Atani, S., Ebrahimi Atani, R. and Tekir, ¨U., ”A Zariski Topology for Semimodules”, European Journal of Pure and Applied Math.,4(3) (2011), 251 - 265.
[6] Ebrahimi Atani, R., ”Prime Subsemimodules of Semimodules”, Int. J.of Algebra, 4(26)
(2010), 1299 - 1306.
[7] Tavallaee, H.A. and Zolfaghari, M., ”Some Remarks on Weakly Prime and Weakly Semiprime Submodules”,Journal of Advanced Research in Pure Math.,4(1) (2012), 19 - 30.
