Abstract

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 and of an -module , the residual of by is defined as the set which is clearly an ideal of . In particular, the ideal is called the annihilator of . Let be a submodule of and let be an ideal of ; the residual submodule of by is defined as . 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 of an -module is a prime submodule if, whenever for and , or . An -module is called a prime module if its zero submodule is a prime submodule. A proper submodule of an -module is called weakly prime (weakly primary) if , where and ; then or ( or ). A proper submodule of an -module is called almost prime (almost primary) if, whenever for and , or ( or ). A proper ideal of a commutative ring is called semiprime if , where and ; then . A proper submodule of an -module is called semiprime if, whenever , , and such that , . An -module is called a second module provided that, for every element , the -endomorphism of produced by multiplication by is either surjective or zero; this implies that is a prime ideal of and is said to be -second [20]. An -module is called a multiplication module provided that, for every submodule of , there exists an ideal of so that (or equivalently, ). An Ideal of a ring is called multiplication if it is multiplication as -modules. Multiplication modules and ideals have been studied extensively in [21–23]. An -module is called a cancellation module if, for all ideals and of , implies that ; see [24]. If is a ring and is an -module, the subset of is defined by for some . Obviously, if is an integral domain, then is a submodule of . If , then we say that is torsion and if , we say that is torsion-free.

2. Almost Semiprime Submodules

Definition 1. (i) Let be a commutative ring. A proper ideal of is called almost semiprime if, whenever for and , .
(ii) Let be a commutative ring and let be an -module. A proper submodule of is called almost semiprime if, whenever , , and such that , .

Let be an -module and let be a submodule of . Following [25], is called idempotent in if . Thus, any proper idempotent submodule of is almost semiprime. If is a multiplication -module and and are two submodules of , then the product of and is defined as ; see [9]. In particular, one has .

Furthermore, is a cancellation -module; then by using Lemma 12, . So in this case, a submodule is idempotent in if and only if . Following [26], a submodule of an -module is called a pure (RD-) submodule if () for any ideal of (for any ). In [25], it was proved that if is a pure submodule in a multiplication -module with pure annihilator, then is idempotent in 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 -module (the integers modulo 24) and the submodule . Then , and so is an almost semiprime submodule of . But is not semiprime in , because , but .

In the semiprime submodules case, is a semiprime submodule of , if and only if is so in for any submodule . But the coverse part may not be true in the case of almost semiprime submodules. For example, for any non-almost semiprime submodule of , we have is an almost semiprime submodule of . For another nontrivial example, we consider the ring , where is a field and ideals , . Then is an almost semiprime submodule of the -module , while is not so in . But we have the following theorem.

Theorem 3. Let and be submodules of an -module with . Then is an almost semiprime submodule of if and only if is an almost semiprime submodule of the -module .

Proof. Let be an almost semiprime submodule of and assume that , , and such that . It is clear that , and so . Therefore since is almost semiprime. Therefore, ; hence is an almost semiprime submodule. Conversely, let be an almost semiprime submodule of and assume that for some , , and . Hence, , because if since , , a contradiction. Therefore , so , as required.

Theorem 4. Let be a multiplicative closed subset of and let be an almost semiprime submodule of -module with . Then is an almost semiprime submodule of the -module .

Proof. Let be an almost semiprime submodule of . Since , then . Assume that , where , , and . Hence, for some and , and so there exists such that . If , then , a contradiction. So , and since is almost semiprime. Therefore, ; hence is an almost semiprime submodule of .

Proposition 5. Let where each is a commutative ring with nonzero identity. Let be an -module and let be the -module with action , where and . Then (i) is an almost semiprime submodule of if and only if is an almost semiprime submodule of ;(ii) is an almost semiprime submodule of if and only if is an almost semiprime submodule of .

Proof. (i) Let be an almost semiprime submodule of . Assume that , where , , and . If , then = , a contradiction. Hence, as is almost semiprime and , then , and so . Conversely, assume that is an almost semiprime submodule of . Let for , , and . Then by (i). Therefore , since is almost semiprime, so , as needed.
(ii) is similar to (i).

Let be a commutative ring with identity and let be an -module. Then with multiplication and with addition is a commutative ring with identity and is a nilpotent ideal of index 2. The ring is said to be the idealization of or trivial extension of by . We view as a subring of via . An ideal is said to be homogeneous if for some ideal of and some submodule of such that .

Lemma 6. Let be an ideal of . Then .

Proof. The proof is straightforward.

Theorem 7. Let be a homogeneous ideal of . Then, if is an almost semiprime ideal of , then is an almost semiprime ideal of and is an almost semiprime submodule of .

Proof. Assume that is an almost semiprime ideal of . Let and such that . Then , because if , then by Lemma 6, , hence , a contradiction. Therefore , and , so is an almost semiprime ideal of .
Let , , and such that . Therefore , because if , then . So since is a homogeneous ideal, a contradiction. Hence , so . Thus, is an almost semiprime submodule of .

Proposition 8. Let be an -module and let be an almost semiprime submodule of . Then(i)if is a second -module, then is a second module;(ii)if is a second -module, then is an -submodule of .

Proof. Let be an almost semiprime submodule of . Let . If , then . Let . Now It is enough to show that . First, we show that . Since is a proper submodule of , for any , we have . Therefore . Let . We may assume that . Since , for some , and for some . Hence , as is almost semiprime so . Hence , so . Therefore and is second.
(ii) Let . If , then , so . Suppose that , so by (i), ; therefore .

In the following theorems, we give other characterizations of almost semiprime submodules.

Theorem 9. Let be an -module and let be a proper submodule of . Then the following are equivalent:(i) is an almost semiprime submodule of .(ii)For and , .(iii)For and , or .

Proof. (i)⇒(ii) Let ; then . If , as is almost semiprime, , so . Let ; then ; hence . The other containment holds for any submodule .
(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 for some , , and . Hence and , so by assumption, and . Therefore is almost semiprime.

The following theorem gives from Theorem 9.

Theorem 10. Let be an -module and let be a proper submodule of . Then is almost semiprime in if and only if for any submodule of , , and with and , one has .

We know that if is a semiprime submodule of , then is a semiprime ideal of . But it may not be true in the case of almost semiprime submodules.

Example 11. Let denote the cyclic -module (the integers modulo 4). Take . Certainly, is almost semiprime, but is not an almost semiprime ideal of , because , but .

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 be a submodule of a finitely generated faithful multiplication (so cancellation) -module. Then for every ideal of .

Proof. The proof is by [12, Lemma 3.4].

Theorem 13. Let be a finitely generated faithful multiplication -module and let be a proper submodule of . Then the following are equivalent.(i) is almost semiprime in .(ii) is almost semiprime in .(iii) for some almost semiprime ideal of .

Proof. (i)⇒(ii) Suppose that is an almost semiprime submodule of . Let and such that . Then and . Indeed, if , then, by Lemma 12, , a contradiction. Now, almost semiprime implies that by Theorem 10, so ; hence is almost semiprime in .
(ii)⇒(i) In this direction, we need to be just a multiplication module. Let , where , , and . Then . Moreover, because, otherwise, if , then , a contradiction. As is an almost semiprime ideal of , . Therefore , and so , as required.
(ii)(iii) We choose .

Lemma 14. Let be a submodule of a faithful multiplication -module and let be a finitely generated faithful multiplication ideal of . Then(i);(ii)if , then for any ideal of .

Proof. It follows from [18].

Theorem 15. Let be a submodule of a faithful multiplication -module and let be a finitely generated faithful multiplication ideal of . Then is an almost semiprime submodule of if and only if is an almost semiprime submodule of .

Proof. Assume that is almost semiprime in . Let , , and such that . Then . In fact, if , then, by Lemma 14, , a contradiction. As is almost semiprime in , then , so ; hence is almost semiprime in .
Conversely, suppose that is almost semiprime in . Let be a submodule of , , and such that . Then . Moreover, if , then, by Lemma 14, , a contradiction. As is almost semiprime in , and so . Therefore is almost semiprime in .

Lemma 16. For every proper ideal of , is an almost semiprime ideal of .

Proof. Since , the proof is held.

Let be a proper submodule of . Then the -radical of , denoted by , is defined to be the intersection of all prime submodules of containing . It is shown in [22] that if is a proper submodule of a multiplication -module , then .

Theorem 17. Let be a finitely generated faithful multiplication -module. Then for every proper submodule of , is an almost semiprime submodule of .

Proof. Let be a proper submodule of . Hence by Lemma 16, is an almost semiprime ideal of . Therefore by Theorem 13, is an almost semiprime submodule of .

3. Weakly Semiprime Submodules

Definition 18. (i) Let be a commutative ring. A proper ideal of is called weakly semiprime if, whenever for some and , .
(ii) Let be an -module. A proper submodule of is called weakly semiprime if, whenever for some , , and , .

Remark 19. Let be a module over a commutative ring . Then semiprime submodules weakly semiprime submodules almost semiprime submodules.

Example 20. Consider the -module and the proper submodule . Then , , and , so . Therefore is almost semiprime. On the other hand, , but , and so is not weakly semiprime.

Theorem 21. Let be an -module and let be a proper submodule of . Then is an almost semiprime submodule of if and only if is a weakly semiprime submodule of the -module .

Proof. Assume that is an almost semiprime submodule of . Let , , and such that . Hence , and so . Therefore , as needed.
Conversely, assume that is weakly semiprime in . Let , where , , and . Then , and hence . Therefore , as required.