We introduce the concept of almost semiprime submodules
of unitary modules over a commutative ring with nonzero identity. We investigate
some basic properties of almost semiprime and weakly semiprime
submodules and give some characterizations of them, especially for (finitely
generated faithful) multiplication modules.
1. Introduction
Throughout this paper, all rings are commutative rings with identity and all modules are unitary. Various generalizations of prime (primary) ideals are studied in [1–8]. The class of prime submodules of modules as a generalization of the class of prime ideals has been studied by many authors; see, for example, [9, 10]. Then many generalizations of prime submodules were studied such as weakly prime (primary) [11], almost prime (primary) [12], 2-absorbing [13], classical prime (primary) [14, 15], and semiprime submodules [16]. In this paper, we study weakly semiprime and almost semiprime submodules as the generalizations of semiprime submodules. Weakly semiprime submodules have been already studied in [17]. Here we first define the notion almost semiprime submodules and get a number of propensities of almost semiprime and weakly semiprime submodules. Also, we give some characterizations of such submodules in multiplication modules. Now we define the concepts that we will use.
For any two submodules N and K of an R-module M, the residual of N by K is defined as the set (N:K)={r∈R:rK⊆N} which is clearly an ideal of R. In particular, the ideal (0:M) is called the annihilator of M. Let N be a submodule of M and let I be an ideal of R; the residual submodule of N by I is defined as (N:MI)={m∈M:Im⊆N}. These two residual ideals and submodules were proved to be useful in studying many concepts of modules; see, for example, [18, 19]. A proper submodule N of an R-module M is a prime submodule if, whenever rm∈N for r∈R and m∈M, m∈N or r∈(N:M). An R-module M is called a prime module if its zero submodule is a prime submodule. A proper submodule N of an R-module M is called weakly prime (weakly primary) if 0≠rm∈N, where r∈R and m∈M; then m∈N or r∈(N:M) (m∈N or r∈rad(N:M)). A proper submodule N of an R-module is called almost prime (almost primary) if, whenever rm∈N-(N:M)N for r∈R and m∈M, m∈N or r∈(N:M) (m∈N or r∈rad(N:M)). A proper ideal I of a commutative ring R is called semiprime if akb∈I, where a,b∈R and k∈Z+; then ab∈I. A proper submodule N of an R-module M is called semiprime if, whenever r∈R, m∈M, and k∈Z+ such that rkm∈N, rm∈N. An R-module M is called a second module provided that, for every element r∈R, the R-endomorphism of M produced by multiplication by r is either surjective or zero; this implies that (0:M)=p is a prime ideal of R and M is said to be p-second [20]. An R-module M is called a multiplication module provided that, for every submodule N of M, there exists an ideal I of R so that N=IM (or equivalently, N=(N:M)M). An Ideal I of a ring R is called multiplication if it is multiplication as R-modules. Multiplication modules and ideals have been studied extensively in [21–23]. An R-module M is called a cancellation module if, for all ideals I and J of R, IM=JM implies that I=J; see [24]. If R is a ring and M is an R-module, the subset T(M) of M is defined by T(M)={m∈M:rm=0 for some 0≠r∈R}. Obviously, if R is an integral domain, then T(M) is a submodule of M. If T(M)=M, then we say that M is torsion and if T(M)=0, we say that M is torsion-free.
2. Almost Semiprime SubmodulesDefinition 1.
(i) Let R be a commutative ring. A proper ideal I of R is called almost semiprime if, whenever akb∈I-I2 for a,b∈R and k∈Z+, ab∈I.
(ii) Let R be a commutative ring and let M be an R-module. A proper submodule N of M is called almost semiprime if, whenever r∈R, m∈M, and k∈Z+ such that rkm∈N-(N:M)N, rm∈N.
Let M be an R-module and let N be a submodule of M. Following [25], N is called idempotent in M if N=(N:M)N. Thus, any proper idempotent submodule of M is almost semiprime. If M is a multiplication R-module and N=IM and K=JM are two submodules of M, then the product NK of N and K is defined as NK=(IM)(JM)=(IJ)M; see [9]. In particular, one has N2=NN=[(N:M)M][(N:M)M]=(N:M)2M.
Furthermore, M is a cancellation R-module; then by using Lemma 12, (N:M)N=((N:M)N:M)M=(N:M)2M=N2. So in this case, a submodule N is idempotent in M if and only if N=N2. Following [26], a submodule N of an R-module M is called a pure (RD-) submodule if IN=N∩IM (rN=N∩rM) for any ideal I of R (for any r∈R). In [25], it was proved that if N is a pure submodule in a multiplication R-module M with pure annihilator, then N is idempotent in M and so is almost semiprime.
Example 2.
(i) It is clear that every semiprime submodule is almost semiprime. But the converse is not true in general. For example, consider Z-module M=Z24 (the integers modulo 24) and the submodule N=〈8〉. Then (N:M)N=N, and so N is an almost semiprime submodule of M. But N is not semiprime in M, because 22.2∈N, but 2.2∉N.
In the semiprime submodules case, N is a semiprime submodule of M, if and only if N/K is so in M/K for any submodule K⊆N. But the coverse part may not be true in the case of almost semiprime submodules. For example, for any non-almost semiprime submodule N of M, we have N/N=0 is an almost semiprime submodule of M/N. For another nontrivial example, we consider the ring R=K[x,y], where K is a field and ideals P=(x,y2), I=(x,y)2. Then P/I is an almost semiprime submodule of the R-module R/I, while P is not so in R. But we have the following theorem.
Theorem 3.
Let N and K be submodules of an R-module M with K⊆(N:M)N. Then N is an almost semiprime submodule of M if and only if N/K is an almost semiprime submodule of the R-module M/K.
Proof.
Let N be an almost semiprime submodule of M and assume that r∈R, m+K∈M/K, and k∈Z+ such that rk(m+K)∈N/K-(N/K:M/K)N/K. It is clear that (N/K:RM/K)=(N:RM), and so rkm∈N-(N:M)N. Therefore rm∈N since N is almost semiprime. Therefore, r(m+k)∈N/K; hence N/K is an almost semiprime submodule. Conversely, let N/K be an almost semiprime submodule of M/K and assume that rkm∈N-(N:M)N for some r∈R, m∈M, and k∈Z+. Hence, rk(m+K)∈N/K-(N/K:M/K)N/K, because if rk(m+K)∈(N/K:M/K)N/K=(N:M)N/K=((N:M)N+K)/K=(N:M)N/K since K⊆(N:M)N, rkm∈(N:M)N, a contradiction. Therefore r(m+K)∈N/K, so rm∈N, as required.
Theorem 4.
Let S be a multiplicative closed subset of R and let N be an almost semiprime submodule of R-module M with S∩(N:M)=∅. Then S-1N is an almost semiprime submodule of the S-1R-module S-1M.
Proof.
Let N be an almost semiprime submodule of M. Since (N:M)∩S=∅, then S-1N≠S-1M. Assume that (r/s)km/t∈S-1N-(S-1:S-1RS-1M)S-1N, where r/s∈S-1R, m/t∈S-1M, and k∈Z+. Hence, rkm/skt=n/s′ for some n∈N and s′∈S, and so there exists t′∈S such that rks′t′m=sktt′n∈N. If rks′t′m∈(N:M)N, then rkm/skt=rks′t′m/skts′t′∈S-1((N:RM)N)=S-1(N:RM)S-1N⊆(S-1N:S-1RS-1M)S-1N, a contradiction. So rks′t′m∈N-(N:M)N, and rs′t′m∈N since N is almost semiprime. Therefore, rm/st=rs′t′m/sts′t′∈S-1N; hence S-1N is an almost semiprime submodule of S-1M.
Proposition 5.
Let R=R1×R2 where each Ri is a commutative ring with nonzero identity. Let Mi be an Ri-module and let M=M1×M2 be the R-module with action (r1,r2)(m1,m2)=(r1m1,r2m2), where ri∈Ri and mi∈Mi. Then
N1 is an almost semiprime submodule of M1 if and only if N1×M2 is an almost semiprime submodule of M;
N2 is an almost semiprime submodule of M2 if and only if M1×N2 is an almost semiprime submodule of M.
Proof.
(i) Let N1 be an almost semiprime submodule of M1. Assume that (r1,r2)k(m1,m2)∈N1×M2-(N1×M2:M)N1×M2, where (r1,r2)∈R, (m1,m2)∈M, and k∈Z+. If r1km1∈(N1:M1)N1, then (r1,r2)k(m1,m2)∈(N1:M1)N1×(M2:M2)M2 = ((N1:M1)×(M2:M2))N1×M2=(N1×M2:M1×M2)N1×M2, a contradiction. Hence, as N1 is almost semiprime and r1km1∈N1-(N1:M1)N1, then r1m1∈N1, and so (r1,r2)(m1,m2)∈N1×M2. Conversely, assume that N1×M2 is an almost semiprime submodule of M. Let r1km1∈N1-(N1:M1)N1 for r1∈R1, m1∈M1, and k∈Z+. Then (r1,1)k(m1,0)∈N1×M2-(N1×M2:M)N1×M2 by (i). Therefore (r1,1)(m1,0)∈N1×M2, since N1×M2 is almost semiprime, so r1m1∈N1, as needed.
(ii) is similar to (i).
Let R be a commutative ring with identity and let M be an R-module. Then R(M)=R(+)M with multiplication (a,m)(b,n)=(ab,an+bm) and with addition (a,m)+(b,n)=(a+b,m+n) is a commutative ring with identity (1,0) and 0(+)M is a nilpotent ideal of index 2. The ring R(+)M is said to be the idealization of M or trivial extension of R by M. We view R as a subring of R(+)M via r→(r,0). An ideal J is said to be homogeneous if J=I(+)N for some ideal I of R and some submodule N of M such that IM⊆N.
Lemma 6.
Let I(+)N be an ideal of R(M). Then (I(+)N)2⊆I2(+)IN.
Proof.
The proof is straightforward.
Theorem 7.
Let I(+)N be a homogeneous ideal of R(M). Then, if I(+)N is an almost semiprime ideal of R(M), then I is an almost semiprime ideal of R and N is an almost semiprime submodule of M.
Proof.
Assume that I(+)N is an almost semiprime ideal of R(M). Let a,b∈R and k∈Z+ such that akb∈I-I2. Then (a,0)k(b,0)∈I(+)N-(I(+)N)2, because if (a,0)k(b,0)∈(I(+)N)2, then by Lemma 6, (a,0)k(b,0)∈I2(+)IN, hence akb∈I2, a contradiction. Therefore (a,0)(b,0)∈I(+)N, and ab∈I, so I is an almost semiprime ideal of R.
Let r∈R, m∈M, and k∈Z+ such that rkm∈N-(N:M)N. Therefore (r,0)k(0,m)∈I(+)N-(I(+)N)2, because if (r,0)k(0,m)=(0,rkm)∈(I(+)N)2⊆I2(+)IN, then rkm∈IN. So rkm∈IN⊆(N:M)N since I(+)N is a homogeneous ideal, a contradiction. Hence (r,0)(0,m)∈I(+)N, so rm∈N. Thus, N is an almost semiprime submodule of M.
Proposition 8.
Let M be an R-module and let N be an almost semiprime submodule of M. Then
if M is a second R-module, then N is a second module;
if M is a second R-module, then N is an RD-submodule of M.
Proof.
Let N be an almost semiprime submodule of M. Let r∈R. If rM=0, then rN⊆rM=0. Let rM=M. Now It is enough to show that N⊆rN. First, we show that (N:M)N=0. Since N is a proper submodule of M, for any r∈(N:M), we have rM=0. Therefore (N:M)N=0. Let n∈N. We may assume that n≠0. Since rM=M, n=rm for some m∈M, and m=rm′ for some m′∈M. Hence n=r2m′∈N-(N:M)N, as N is almost semiprime so m=rm∈N. Hence n=rm∈rN, so N⊆rN. Therefore rN=N and N is second.
(ii) Let r∈R. If rM=0, then rN=0, so rN=0=N∩rM. Suppose that rM=M, so by (i), rN=N; therefore rN=N∩rM.
In the following theorems, we give other characterizations of almost semiprime submodules.
Theorem 9.
Let M be an R-module and let N be a proper submodule of M. Then the following are equivalent:
N is an almost semiprime submodule of M.
For r∈R and k∈Z+, (N:M〈rk〉)=(N:M〈r〉)∪((N:M)N:M〈rk〉).
For r∈R and k∈Z+, (N:M〈rk〉)=(N:M〈r〉) or (N:M〈rk〉)=((N:M)N:M〈rk〉).
Proof.
(i)⇒(ii) Let m∈(N:M〈rk〉); then rkm∈N. If rkm∉(N:RM)N, as N is almost semiprime, rm∈N, so m∈(N:M〈r〉). Let rkm∈(N:RM)N; then m∈((N:RM)N:M〈r〉k); hence (N:M〈rk〉)⊆(N:M〈r〉)∪((N:M)N:M〈rk〉). The other containment holds for any submodule N.
(ii)⇒(iii) It is well known that if a submodule is the union of two submodules, then it is equal to one of them.
(iii)⇒(i) Let rkm∈N-(N:RM)N for some r∈R, m∈M, and k∈Z+. Hence m∈(N:M〈rk〉) and m∉((N:RM)N:M〈rk〉), so by assumption, m∈(N:M〈r〉) and rm∈N. Therefore N is almost semiprime.
The following theorem gives from Theorem 9.
Theorem 10.
Let M be an R-module and let N be a proper submodule of M. Then N is almost semiprime in M if and only if for any submodule K of M, a∈R, and k∈Z+ with 〈a〉kK⊆N and 〈a〉kK⊈(N:RM)N, one has 〈a〉K⊆N.
We know that if N is a semiprime submodule of M, then (N:RM) is a semiprime ideal of R. But it may not be true in the case of almost semiprime submodules.
Example 11.
Let M denote the cyclic Z-module Z4 (the integers modulo 4). Take N={0}. Certainly, N is almost semiprime, but (N:RM)=4Z is not an almost semiprime ideal of Z, because 22∈(N:M)-(N:M)2, but 2∉(N:M).
Now in the following theorem, we give a characterization of almost semiprime submodules in (finitely generated faithful) multiplication modules. We first need the following lemma.
Lemma 12.
Let N be a submodule of a finitely generated faithful multiplication (so cancellation) R-module. Then (IN:M)=I(N:M) for every ideal I of R.
Proof.
The proof is by [12, Lemma 3.4].
Theorem 13.
Let M be a finitely generated faithful multiplication R-module and let N be a proper submodule of M. Then the following are equivalent.
N is almost semiprime in M.
(N:RM) is almost semiprime in R.
N=PM for some almost semiprime ideal P of R.
Proof.
(i)⇒(ii) Suppose that N is an almost semiprime submodule of M. Let a,b∈R and k∈Z+ such that akb∈(N:M)-(N:M)2. Then 〈a〉k(bM)⊆N and 〈a〉k(bM)⊈(N:M)N. Indeed, if 〈a〉k(bM)⊆(N:M)N, then, by Lemma 12, akb∈((N:M)N:M)=(N:M)2, a contradiction. Now, N almost semiprime implies that 〈a〉(bM)⊆N by Theorem 10, so ab∈(N:M); hence (N:M) is almost semiprime in R.
(ii)⇒(i) In this direction, we need M to be just a multiplication module. Let rkm∈N-(N:M)M, where r∈R, m∈M, and k∈Z+. Then 〈r〉k(〈m〉:M)⊆(〈rkm〉:M)⊆(N:M). Moreover, 〈r〉k(〈m〉:M)⊈(N:M)2 because, otherwise, if 〈r〉k(〈m〉:M)⊆(N:M)2⊆((N:M)N:M), then 〈r〉k〈m〉=〈r〉k(〈m〉:M)M⊆(N:M)N, a contradiction. As (N:M) is an almost semiprime ideal of R, 〈r〉(〈m〉:M)⊆(N:M). Therefore 〈r〉〈m〉=〈r〉(〈m〉:M)M⊆(N:M)M=N, and so rm∈N, as required.
(ii)⇔(iii) We choose P=(N:M).
Lemma 14.
Let N be a submodule of a faithful multiplication R-module M and let I be a finitely generated faithful multiplication ideal of R. Then
N=(IN:MI);
if N⊆IM, then (JN:MI)=J(N:MI) for any ideal J of R.
Proof.
It follows from [18].
Theorem 15.
Let N be a submodule of a faithful multiplication R-module M and let I be a finitely generated faithful multiplication ideal of R. Then N is an almost semiprime submodule of IM if and only if (N:MI) is an almost semiprime submodule of M.
Proof.
Assume that N is almost semiprime in IM. Let r∈R, m∈M, and k∈Z+ such that rkm∈(N:MI)-((N:MI):RM)(N:MI). Then 〈r〉kIm⊆N-(N:RIM)N. In fact, if 〈r〉k(Im)⊆(N:IM)N, then, by Lemma 14, rkm∈((N:IM)N:MI)=(N:IM)(N:MI)=((N:MI):M)(N:MI), a contradiction. As N is almost semiprime in IM, then 〈r〉(Im)⊆N, so rm∈(N:MI); hence (N:MI) is almost semiprime in M.
Conversely, suppose that (N:MI) is almost semiprime in M. Let K be a submodule of IM, a∈R, and k∈Z+ such that 〈a〉nK⊆N-(N:M)N. Then 〈a〉n(K:MI)⊆(〈a〉nK:MI)⊆(N:MI). Moreover, if 〈a〉n(K:MI)⊆((N:MI):M)(N:MI)=(N:RIM)(N:MI), then, by Lemma 14, 〈a〉nK=〈a〉n(IK:MI)=〈a〉n(K:MI)I⊆(N:IM)(N:MI)I=(N:IM)N, a contradiction. As (N:MI) is almost semiprime in M, 〈a〉(K:MI)⊆(N:MI) and so 〈a〉K=〈a〉(K:MI)⊆(N:MI)I=N. Therefore N is almost semiprime in IM.
Lemma 16.
For every proper ideal I of R,
rad
(I) is an almost semiprime ideal of R.
Proof.
Since (rad(I))2=rad(I), the proof is held.
Let N be a proper submodule of M. Then the M-radical of N, denoted by M-rad(N), is defined to be the intersection of all prime submodules of M containing N. It is shown in [22] that if N is a proper submodule of a multiplication R-module M, then M-rad(N)=rad(N:RM)M.
Theorem 17.
Let M be a finitely generated faithful multiplication R-module. Then for every proper submodule N of M, M-rad(N) is an almost semiprime submodule of M.
Proof.
Let N be a proper submodule of M. Hence by Lemma 16, rad(N:RM) is an almost semiprime ideal of R. Therefore by Theorem 13, M-rad(N)=rad(N:RM)M is an almost semiprime submodule of M.
3. Weakly Semiprime SubmodulesDefinition 18.
(i) Let R be a commutative ring. A proper ideal I of R is called weakly semiprime if, whenever 0≠akb∈I for some a,b∈R and k∈Z+, ab∈I.
(ii) Let M be an R-module. A proper submodule N of M is called weakly semiprime if, whenever 0≠rkm∈N for some r∈M, m∈M, and k∈Z+, rm∈N.
Remark 19.
Let M be a module over a commutative ring R. Then semiprime submodules ⇒ weakly semiprime submodules ⇒ almost semiprime submodules.
Example 20.
Consider the Z-module M=Z24 and the proper submodule N=〈8〉={0,8,16}. Then 0=0.8, 8=16.8, and 16=16.16, so (N:M)N=N. Therefore N is almost semiprime. On the other hand, 0≠22.2∈N, but 2.2∉N, and so N is not weakly semiprime.
Theorem 21.
Let M be an R-module and let N be a proper submodule of M. Then N is an almost semiprime submodule of M if and only if N/(N:M)N is a weakly semiprime submodule of the R-module M/(N:M)N.
Proof.
Assume that N is an almost semiprime submodule of M. Let r∈R, m+(N:M)N∈M/(N:M)N, and k∈Z+ such that 0≠rk(m+(N:M)N∈N/(N:M)N. Hence rkm∈N-(N:M)N, and so rm∈N. Therefore r(m+(N:M)N)∈N/(N:M)N, as needed.
Conversely, assume that N/(N:M)N is weakly semiprime in M/(N:M)N. Let rkm∈N-(N:M)N, where r∈R, m∈M, and k∈Z+. Then 0≠rk(m+(N:M)N)∈N/(N:M)N, and hence r(m+(N:M)N)∈N/(N:M)N. Therefore rm∈N, as required.
Proposition 22.
Let R be an integral domain and let M be a torsion-free R-module. Then every weakly semiprime submodule of M is semiprime.
Proof.
Let N be a weakly semiprime submodule of M. Let r∈R, m∈M, and k∈Z+ such that rkm∈N. If 0≠rkm, then N weakly semiprime gives that rm∈N. Suppose that rkm=0. If rk≠0, then m∈T(M)=0, so rm∈N. If rk=0, then r=0, and hence rm∈N. Therefore N is semiprime.
Proposition 23.
Let M be a prime R-module. Then every weakly semiprime submodule of M is semiprime.
Proof.
Let N be a weakly semiprime submodule of M. Let r∈R, m∈M, and k∈Z+ such that rkm∈N. If 0≠rkm, then N weakly semiprime gives that rm∈N. Suppose that rkm=0; then rm=0 or rk-1M=0 since M is a prime module. By following this method, we get rm=0∈N; hence N is a semiprime submodule of M.
Proposition 24.
Let M be a second R-module and let N be a proper submodule of M. Then N is almost semiprime if and only if N is weakly semiprime.
Proof.
We know that every weakly semiprime is almost semiprime. Let N be an almost semiprime submodule of M and 0≠rkm∈N for some r∈R, m∈M, and k∈Z+. By Proposition 8, we have (N:M)N=0; hence rkm∈N-(N:M)N, and so rm∈N. Therefore N is weakly semiprime submodule of M.
Now we get other characterizations of weakly semiprime submodule.
Theorem 25.
Let M be an R-module and let N be a proper submodule of M. Then the following are equivalent.
N is a weakly semiprime submodule of M.
For r∈R and k∈Z+, (N:M〈rk〉)=(0:M〈rk〉)∪(N:M〈r〉).
For r∈R and k∈Z+, (N:M〈rk〉)=(0:M〈rk〉) or (N:M〈rk〉)=(N:M〈r〉).
Proof.
(i)⇒(ii) Let m∈(N:M〈rk〉); then rkm∈N. If rkm≠0, as N is weakly semiprime, rm∈N, so m∈(N:M〈r〉). Let rkm=0; then m∈(0:M〈rk〉), and hence (N:M〈rk〉)⊆(N:M〈r〉)∪(0:M〈rk〉). Clearly, (N:M〈r〉)∪(0:M〈rk〉)⊆(N:M〈rk〉); therefore (N:M〈rk〉)=(0:M〈rk〉)∪(N:M〈r〉).
(ii)⇒(iii) It is straightforward.
(iii)⇒(i) Let 0≠rkm∈N for some r∈R, m∈M, and k∈Z+. Hence m∈(N:M〈rk〉) and m∉(0:M〈rk〉), so by assumption, m∈(N:M〈r〉). Therefore N is weakly semiprime.
Theorem 26.
Let M be an R-module and let N be a proper submodule of M. Then N is weakly semiprime in M if and only if for any submodule K of M, a∈R, and k∈Z+ with 0≠〈a〉kK⊆N, one has 〈a〉K⊆N.
Theorem 27.
Let N be a weakly semiprime submodule of an R-module M with T(M)=0. Then for any nonzero ideal I of R, (N:MI) is a weakly semiprime submodule of M.
Proof.
Let r∈R, m∈M, and k∈Z+ such that 0≠rkm∈(N:MI). Hence 〈r〉k(mI)⊆N. If 0≠〈r〉k(mI)⊆N, then, by Theorem 26, N weakly semiprime gives that 〈r〉(mI)∈N, so rm∈(N:MI), as needed. Suppose that 〈r〉k(mI)=0, so rkma=0 for some nonzero a∈I. Hence rkm∈T(M)=0, which is a contradiction. Therefore (N:MI) is weakly semiprime.
In Theorem 27, the assumption T(M)=0 is necessary. To see this, consider Z-module Z16. Let N={0} and I=2Z. Clearly, N is weakly semiprime submodule of M, but (N:MI)={0,8} is not weakly semiprime.
Theorem 28.
Let I be an ideal of R and let N be a submodule of M such that I(+)N is a weakly semiprime ideal of R(M). Then I is a weakly semiprime ideal of R and N is a weakly semiprime submodule of M.
Proof.
Assume that I(+)N is a weakly semiprime ideal of R(M). Let a,b∈R and k∈Z+ such that 0≠akb∈I. Then (0,0)≠(a,0)k(b,0)∈I(+)N. Therefore (a,0)(b,0)∈I(+)N, and ab∈I, so I is a weakly semiprime ideal of R.
Let r∈R, m∈M, and k∈Z+ such that 0≠rkm∈N. Therefore (0,0)≠(r,0)k(0,m)∈I(+)N; hence (r,0)(0,m)∈I(+)N, so rm∈N. Thus, N is a weakly semiprime submodule of M.
In [17], the authors have proved that if N is a weakly semiprime submodule of a faithful cyclic R-module M, then the ideal (N:M) is weakly semiprime. But we show that the assumption faithful R-module for this theorem is sufficient.
Proposition 29.
Let M be a faithful R-module and let N be a weakly semiprime submodule of M. Then (N:M) is a weakly semiprime ideal of R.
Proof.
Suppose N is weakly semiprime, a,b∈R, and k∈Z+ such that 0≠akb∈(N:M). Then 0≠〈a〉k(bM)⊆N. Indeed, if 〈a〉k(bM)=0, then akb∈(0:M)=0, a contradiction. Now, Theorem 25 implies that 〈a〉(bM)⊆N, so ab∈(N:M), and (N:M) is weakly semiprime in R.
Now we give characterizations of weakly semiprime submodules in (finitely generated faithful) multiplication modules.
Theorem 30.
Let M be a finitely generated faithful multiplication R-module and let N be a proper submodule of M. Then the following are equivalent.
N is weakly semiprime in M.
(N:RM) is weakly semiprime in R.
N=QM for some weakly semiprime ideal Q of R.
Proof.
(i)⇒(ii) It follows from Proposition 29.
(ii)⇒(i) In this direction, we need M to be just a multiplication module. Let 0≠rkm∈N, where r∈R, m∈M, and k∈Z+. Then 〈r〉k(〈m〉:M)⊆(〈rkm〉:M)⊆(N:M). Moreover, 〈r〉k(〈m〉:M)≠0 because, otherwise, if 〈r〉k(〈m〉:M)=0, then 〈r〉k〈m〉=〈r〉k(〈m〉:M)M=0, a contradiction. As (N:M) is a weakly semiprime ideal of R, 〈r〉(〈m〉:M)⊆(N:M). Therefore 〈r〉〈m〉=〈r〉(〈m〉:M)M⊆(N:M)M=N, and so rm∈N, as required.
(ii)⇔(iii) We choose Q=(N:M).
Definition 31.
A proper ideal I of an integral domain R is said to be divided if I⊂Rx for all x∈R-I (see [27]). Generalizing this idea to modules one says that a proper submodule N of an R-module M is divided if N⊂Rm for all m∈M-N.
Lemma 32.
Let R be a commutative ring and let M be a finitely generated faithful multiplication R-module. If P is a divided prime submodule of M, then (P:M) is a divided prime ideal of R.
Proof.
The proof is by [28, Proposition 6].
Theorem 33.
Let R be a commutative ring, let M be a finitely generated faithful multiplication R-module, and let N be a proper submodule of M such that M-
rad
(N)=P, where P is a divided prime submodule of M. Then
if N is a semiprime submodule of M, then N is a primary submodule of M;
if N is an almost semiprime submodule of M, then N is an almost primary submodule of M;
if N is a weakly semiprime submodule of M, then N is a weakly primary submodule of M.
Proof.
Note first that P=(P:M)M, where (P:M)=p is a prime ideal of R. Also, M-rad(N)=rad(N:M)M. On the other hand, M-rad(N)=P=pM=rad(N:M)M. Moreover, every finitely generated faithful multiplication module is a cancellation, so that p=(P:M)=rad(N:M).
(i) Assume that am∈N and a∉rad((N:M))=p. Then from am∈P and a∉p, we get m∈P since P is prime. By Lemma 32, p is the divided prime ideal of R. So p⊂Ra since a∉p. Therefore P=pM⊂Ma, and hence m=am′ for some m′∈M. Now it follows from am=a2m′∈N, and so m=am′∈N since N is assumed to be semiprime submodule. This shows that N is a primary submodule of M.
The proofs of (ii) and (iii) follow from (i).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research of the author is supported by a grant from Payame Noor University (PNU). The author would like to thank the referee(s) for valuable comments and suggestions which have improved the paper.
AndersonD. D.BatainehM.Generalizations of prime ideals200836268669610.1080/00927870701724177MR2388034ZBL1140.130052-s2.0-41849084153AndersonD. D.SmithE.Weakly prime ideals2003294831840MR2045656ZBL1086.135002-s2.0-0344118036AndersonD. F.BadawiA.On n-absorbing ideals of commutative rings20113951646167210.1080/00927871003738998MR2821499ZBL1232.130012-s2.0-79959523749BadawiA.On 2-absorbing ideals of commutative rings200775341742910.1017/S0004972700039344MR2331019ZBL1120.130042-s2.0-34547275279BadawiA.DaraniA. Y.On weakly 2-absorbing ideals of commutative rings2013392441452MR30804482-s2.0-84881422675Ebrahimi AtaniS.FarzalipourF.On weakly primary ideals2005123423429MR2174944EbrahimpourM.NekooeiR.On generalizations of prime ideals20124041268127910.1080/00927872.2010.550794MR2912983ZBL1278.130032-s2.0-84859880537Yousefian DaraniA.PuczyłowskiE. R.On 2-absorbing commutative semigroups and their applications to rings2013861839110.1007/s00233-012-9417-zMR30162632-s2.0-84873525473AmeriR.On the prime submodules of multiplication modules20032003271715172410.1155/S0161171203202180MR19810262-s2.0-12844252949LuC. P.Prime submodules of modules19843316169MR741378Ebrahimi AtaniS.FarzalipourF.On weakly prime submodules200731371378KhashanH. A.On almost prime submodules201232264565110.1016/S0252-9602(12)60045-9MR29219052-s2.0-84857730899DaraniA. Y.SoheilniaF.2-absorbing and weakly 2-absorbing submodules201193577584MR28620512-s2.0-83255175716BaziarM.BehboodiM.Classical primary submodules and decomposition theory of modules20098335136210.1142/S0219498809003369MR25359942-s2.0-67649086105BehboodiM.2004Ahvaz, IranChamran UniversitySaraçB.On semiprime submodules20093772485249510.1080/00927870802101994MR25369362-s2.0-70449494863TavallaeeH. A.ZolfaghariM.Some remarks on weakly prime and weakly semiprime submodules201241193010.5373/jarpm.745.012311MR2890864AliM.Residual submodules of multiplication modules2004456174NaoumA. G.HasanM. A. K.The residual of finitely generated multiplication modules198646322523010.1007/BF01194187MR8348402-s2.0-1642409694YasemiS.The dual notion of prime submodules200137273278AndersonD. D.Some remarks on multiplication ideals II20002852577258310.1080/00927870008826980MR17574832-s2.0-0034346995El-BastZ. A.SmithP. F.Multiplication modules198816475577910.1080/00927878808823601MR932633SmithP. F.Some remarks on multiplication modules198850322323510.1007/BF01187738MR933916ZBL0615.130032-s2.0-0000460776AndersonD. D.Cancellation modules and related modules2001220New York, NY, USAMarcel Dekker1325Lecture Notes in Pure and Applied MathematicsAliM. M.SmithD. J.Pure submodules of multiplication modules20054516174MR20706332-s2.0-2942677203AndersonF. W.FullerK. R.197413New York, NY, USASpringerGraduate Texts in MathematicsMR1245487BadawiA.On divided commutative rings19992731465147410.1080/00927879908826507MR1669131ZBL0923.130012-s2.0-0033240684AliM. M.Invertibility of multiplication modules III20093919321310.1080/03014220909510578MR2772409