Some Properties of -Semiannihilator Small Submodules and -Small Submodules with respect to a Submodule

Abstract

Let be a commutative ring with nonzero identity, be a multiplicatively closed subset of , and be a unital -module. In this article, we introduce the concepts of -semiannihilator small submodules and --small submodules as generalizations of -small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (finitely generated faithful) multiplication modules.

1. Introduction

Throughout this paper, is a commutative ring with nonzero identity and denotes a unital -module. Also, is a multiplicatively closed subset of . We use the notations “” and “” to denote inclusion and submodules, respectively. As usual, the rings of natural numbers, integers, and integer modulo will be denoted by , , and , respectively. A module over a ring (not necessarily commutative) is called prime if for every nonzero submodule of , . An -module is called faithful if . An -module is called a multiplication module, if for any submodule of , for some ideal of , and in this case, (see [1]). A submodule of an -module is called small (superfluous) which is denoted by , if for any submodule of , , which implies that . It is clear that the zero submodule of every nonzero module is small. More details about small submodules can be found in [2, 3]. In [4], the author introduced the concept of a semiannihilator small submodule of a module over a commutative ring such that is called semiannihilator small (sa-small for short), denoted by , if for every submodule of with implies that . An ideal of is an sa-small ideal of if it is an sa-small submodule of as an -module. Let be an arbitrary submodule of . In [5], a submodule is called a -small submodule of provided for each submodule of , , which implies that .

A nonempty subset of is called a multiplicatively closed subset of if , , and for all , see [6]. Let be an -module and be a multiplicatively closed subset of . Then, is called an -multiplication module if for each submodule of , there exist and an ideal of such that [7]. The concept of -Noetherian rings has been introduced and investigated by Anderson et al. [8]. Farshadifar introduced and studied in brief the notions of -secondary submodules and -copure submodules [9, 10]. Şengelen Sevim et al. in [11] described the concept of -prime submodules. After, the generalizations of -prime submodules have been studied in [12, 13]. Recently, the concept of -small submodules have been studied in [14]. Here, we introduce and study the notions of -semiannihilator small submodules and --small submodules as generalizations of -small submodules. In Sections 2 and 3, various properties of such submodules are considered.

2. -Semiannihilator Small Submodules

In this section, we define the concept of -semiannihilator small submodules of an -module and we get some characterizations of them.

We begin with the following definition.

Definition 1. Let be an -module.(1)We say that a submodule of is an -small submodule of which is denoted by if there exists such that whenever for some submodule of , it implies that . We say that an -module is an -hollow module if every submodule of is an -small submodule of .(2)An ideal of is called -small if it is an -small submodule of as an -module. A ring is an -hollow ring if it is an -hollow -module.

Remark 2. The following results follow from the definition:(1)Clearly, if , then . Particularly, if is an -module with , then is -hollow. Moreover, in this case, is an -multiplication -module because suppose that for some . It is sufficient to take ; then, for every submodule of , as needed.(2)Let be an -module and . Then, there exists a proper submodule of such that . If , then there exists such that . This implies that and so .(3)It is clear that every small submodule is also -small. In particular, the zero submodule is an -small submodule of . The following example shows that converse is not necessarily true in general. Clearly, if and is an -small submodule of , then is small.(4)If , then for every submodule of with , , since there exists such that , as needed.

Example 1. (1)Consider as a -module and take . Then, is not an -small submodule of because for but for all .(2)Consider the -module and the submodule . Take the multiplicatively closed subset . Then, is an -small submodule of . Because we have and , let . Then, and . But is not a small submodule of because and . In general, let be distinct prime numbers and consider the -module . Then, the submodule is an -small submodule of such that . Moreover, the submodule is an -small submodule of such that .

Example 2. Consider as a -module such that is a prime number. It is clear that every proper submodule of is prime, and for any submodule of , . Also, is a prime -module, and it is an -hollow module such that .

Proposition 3. Let be an -module. Then, the following statements are true:(1)If and , then .(2)Let be a nonempty set of -small submodules of . Then, is an -small submodule.

Proof. The proofs are straightforward.
We recall that a submodule of an -module is a semiannihilator small (briefly, sa-small) submodule if whenever for some submodule of , implying that .

Definition 4. (1)A submodule of is called an -semiannihilator small (briefly, -sa-small) submodule of which is denoted by if there exists such that whenever for some submodule of , it implies that .(2)An ideal of is called -semiannihilator small (briefly, -sa-small) ideal if it is an -semiannihilator small submodule of as an -module.

Example 3. Consider the -module and the submodule . Take the multiplicatively closed subset . Then, is an -sa-small submodule of . Because we have and , let . Then, and . But is not an sa-small submodule of because , but .

Lemma 5. Let be an -module and be two multiplicatively closed subsets of with . If is an -sa-small submodule of , then is a -sa-small submodule of .

Proof. The proof is straightforward.
Let be a multiplicatively closed subset of . The saturation of is the set . It is clear that is a multiplicatively closed subset of and that .

Proposition 6. Let be a submodule of an -module . Then, is an -sa-small submodule of if and only if is an -sa-small submodule of .

Proof. Let be an -sa-small submodule of . Then, by Lemma 5, is an -sa-small submodule of . Conversely, let be an -sa-small submodule of . Suppose for some submodule of . Then, there exists such that . Then, there exists such that . We have and so .

Proposition 7. Let be an -module and be submodules of . Then, the following assertions hold:(i)If and , then (ii)If , then

Proof. (i)It is clear.(ii)Let for some submodule of . Then, , so (modular law). Hence, there exists such that . Therefore, .

Proposition 8. Let be an -module and be an ideal of . Then, the following assertions hold:(i)If , then (ii)If is a finitely generated faithful multiplication module and , then

Proof. (i)Let for some ideal of . Then, . Hence, there exists such that , since , so .(ii)Let for some submodule of . We have for some ideal of . Thus, . By Nakayama’s lemma, there exists such that . Since is faithful, and so . Thus, . Since , there exists such that , as needed.

Theorem 9. Let and be -modules and be an -epimorphism. If , then .

Proof. Let for some submodule of . Since is an epimorphism, we have . Hence, , so there exists such that . Thus, since . Therefore, .
By the following example, we show that if is an epimorphism, then the image of an -sa-small submodule of need not be -sa-small in .

Example 4. Consider the -modules and , the multiplicatively closed subset , and the natural epimorphism . Then, is an -sa-small submodule of , but is not an -sa-small submodule of .

The following example shows that the sum of -sa-small submodules of an -module need not be an -sa-small submodule of .

Example 5. Consider the -module and the multiplicatively closed subset . The submodules and are the -sa-small submodules of . But is not an -sa-small submodule of .

Proposition 10. Let and be -modules. If and , then .

Proof. Let for be the projection maps. Since and , by Theorem 9, and . Hence, by Proposition 7.

Definition 11. An -module is called an -semiannihilator hollow (briefly, -sa-hollow) module if every proper submodule of is an -sa-small submodule of .

Example 6. (1)Consider the -module and the multiplicatively closed subset . Then, is an -sa-hollow module, but it is not an -hollow module. Because , but for any , .(2)Consider the -module and the multiplicatively closed subset . Then, is not an -sa-hollow module. Because , but for any , .

Proposition 12. Let be -modules and be an epimorphism. If is an -sa-hollow module, then is an -sa-hollow module.

Proof. Let be a submodule of . Then, is a submodule of . Since is an -sa-hollow module, then . Thus, , and since , by Proposition 7, . Therefore, is an -sa-hollow module.

Example 7. We consider the and as -modules, the multiplicatively closed subset , and the natural epimorphism . Then, is an -sa-hollow -module, and ; is not an -sa-small submodule of . Because , but for any , .

Corollary 13. Let be an -module and be a submodule of . If is an -sa-hollow module, then is an -sa-hollow module.

Proof. Apply Proposition 12.

Theorem 14. Let be an -module and be a submodule of . Assume that is a faithfully flat -module. Then, if is an -sa-small submodule of -module , then is an -sa-small submodule of .

Proof. Let and for some submodule of . Then,By assumption, there exists such that . Since is a faithfully flat -module, . Thus, , so is an -sa-small submodule of .

3. -Small Submodules with respect to a Submodule

Let be an arbitrary submodule of an -module and be a multiplicatively closed subset of . In this section, we introduce and study another generalization of -small and -small submodules, namely, --small submodules.

Definition 15. (1)Let be an -module and be an arbitrary submodule of . A submodule of is called an --small submodule of which is denoted by if there exists such that whenever for some submodule of , it implies that . We say that is an --hollow module if every submodule of is --small in .(2)Let be an ideal of . An ideal of is called --ideal of if there exists such that whenever for some ideal of , then . is an --hollow ring if it is an --hollow as an -module.

Observation 16. Let be an -module.(1)Take ; then, if and only if . If , then every submodule of is --small in .(2)Clearly, every --small submodule is a -small submodule, but the following example shows that converse is not necessarily true.(3)If and , then there exists such that , so . Equivalently, if for some submodule of , , then either or .

Example 8. Consider the -module , the multiplicatively closed subset , and the submodule . Then, the submodule is not a -small submodule of because , but . While, is an -small submodule of . Let be an integer such that . Set . Thus, .

Proposition 17. Let be an -module, , and . Then, the following assertions hold:(i)If , then (ii) if and only if

Proof. (i)Let for some submodule of . We show that for some . We have . Since , there exists such that . Thus, , so .(ii)Suppose that and for some submodule of . Since and , so there exists such that . Thus, . Conversely, let and for some . Then, , and hence, for some because . Thus, for some , so .

Proposition 18. Let be an -module with submodules and . Then, if and only if .

Proof.   It is clear.  Let for some submodule of . Then, . Since , there exists such that .

Theorem 19. Let be an -module with submodules and . If , then and .

Proof. Let for some submodule of . Thus, , so there exists such that . Therefore, . Let for some submodule of . Hence, , so there exists such that since . Thus, .

Definition 20. Let be -modules and . An -epimorphism is called -small in case is an --small submodule of .

Proposition 21. Let be an -module and and be submodules of . The following statements are equivalent:(i)(ii)The natural map is -small(iii)For every -module and -homomorphism , implies that for some

Proof.   and are clear. . Let for some submodule of . Let be the inclusion map. Then, , and by , there exists such that .

Proposition 22. Let be -modules and be an -homomorphism. If and are submodules of such that , then . In particular, if , then .

Proof. Let for some submodule of . It is easy to see that . Thus, there exists such that . Hence, , so . Therefore, .

Corollary 23. Let be -modules and be an -monomorphism. If and are submodules of , then if and only if .

Proof.   By Proposition 22.  Let for some submodule of . Then, . Thus, there exists such that , so . Hence, since is a monomorphism. Therefore, is an -small submodule of .

Example 9. Consider the -homomorphism with and the multiplicatively closed subset . Then, the submodule is an -small submodule of , but is not -small in . Because , for every .

Theorem 23. Let be a Noetherian -module and be a submodule of . Then, is an --small submodule of if and only if is an -small submodule of .

Proof.   Let be an --small submodule of and for some submodule of . Then, we have since is a Noetherian -module. Hence, there exists such that . Let . Then, for some and , so . Therefore, .  It is obvious.

Proposition 24. Let be -modules and and . If is an --small submodule of and is an --small submodule of , then is an --small submodule of .

Proof. Let for some submodule of . Then, and . Thus, there exist such that and . Set . Hence, .

Theorem 25. Let be a finitely generated faithful multiplication -module and . Then, is an --small submodule of if and only if is an --small ideal of .

Proof. Let and for some ideal of . Then, , and since is a multiplication module, we have . Hence, there exists such that , so . Since is a cancellation module, . Therefore, is an --small ideal of . Conversely, let and for some . Thus, since is a multiplication module, so since is a cancellation module. By assumption, there exists such that , and hence, . This implies that is an --small submodule of .

Theorem 26. Let be an -module and . Assume that is a faithfully flat -module. If is an --small submodule of , then is an --small submodule of .

Proof. Let and for some submodule of . Then,Hence, there exists such that . Thus, is exact, so is exact since is faithfully flat. Thus, and , as needed.

4. Conclusions

In this article, we introduced the concepts of -semiannihilator small submodules and --small submodules as generalizations of -small submodules. We showed the concepts of annihilator small submodules and -small submodules are different from the concept of -small submodules. Several properties, examples, and characterizations of such submodules have been investigated. Moreover, we investigated the properties and the behavior of these structures under homomorphisms, Cartesian product, and localizations.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.