Abstract

This article introduces the concept of -semiprime submodules which are a generalization of semiprime submodules and -prime submodules. Let be a nonzero unital R-module, where is a commutative ring with a nonzero identity. Suppose that is a multiplicatively closed subset of . A submodule of is said to be an -semiprime submodule if there exists a fixed , and whenever for some , and , then . Also, is said to be an -reduced module if there exists (fixed) , and whenever for some , and then . In addition, to give many examples and characterizations of -semiprime submodules and -reduced modules, we characterize a certain class of semiprime submodules and reduced modules in terms of these concepts.

1. Introduction

In this article, all rings are assumed to be commutative with a nonzero identity, and all modules are assumed to be nonzero unital. Let always denote such a ring and always denote such an -module. Recalling from [1], an -module is said to be a reduced module if for each and implying that . Note that is a reduced module if and only if for some , and implying that Let be a submodule of is said to be a semiprime submodule; if where and then [2]. It is easy to see that is a reduced module if and only if the zero submodule is semiprime. Also, it is clear that a submodule of is semiprime if and only if for some , and implying that . As a generalization of the prime submodule (torsion-free module), the notion of the semiprime submodule (reduced module) has been widely studied in many papers. See, for example, [1–5]. Our aim, in this paper, is to introduce -semiprime submodules and -reduced modules which are generalizations of semiprime submodules and reduced modules, respectively. For the sake of completeness, we begin by giving some notions and notations which will be used throughout the paper. and denote the set of all prime ideals and maximal ideals of , respectively. A nonempty subset of is said to be a multiplicatively closed set (briefly, m.c.s) if and is a subsemigroup of under multiplication. Let be a submodule of be a nonempty subset of , and be a nonempty subset of ; the residuals of by and are defined as follows:

In particular, if is the singleton, where , we use instead of , and also, we use to denote . For each the set is denoted by , and if is called a zero divisor on . Furthermore, the set of all zero divisors on is denoted by . It is clear that the set of all units of and are always a m.c.s of .

The concepts of prime ideals/submodules have a distinguished place in commutative algebra. Since certain class of rings and modules are characterized in terms of prime ideals/submodules, they have been widely studied by many authors. See, for example, [6–11]. Recently, Sevim et al., in [12], introduced -prime submodules and -torsion-free modules and used them to characterize certain prime submodules and torsion-free modules. Let be a m.c.s of . A submodule of is said to be an -prime submodule if there exists a fixed , and whenever then either or for each and . Note that if then is (trivially) an -prime submodule, and so, the authors in [12] defined -prime submodules with the condition that to avoid the trivial case. Similarly, an -module with is said to be an -torsion-free module if there exists a fixed , and whenever for some and then either or . They showed in [12], Theorem 2.26, that -module with is a simple module if and only if its each proper submodule is an -prime submodule.

Let be a m.c.s of and be a submodule of . Then, we call an -semiprime submodule if there exists a fixed , and whenever for some , and then . Also, is said to be an -reduced module if there exists (fixed) , and whenever for some , and then . To avoid the trivial case, we assume that for each -semiprime submodule of and for each -reduced module . Among other results in this paper, we show that the class of -semiprime submodules properly contains the class of semiprime submodules and the class of -prime submodules (see Proposition 1 and Examples 1 and 2). Also, we show that, in Proposition 2, if is an -semiprime submodule of then is a semiprime submodule of the quotient module of . Also, we investigate the behaviour of -semiprime submodules under homomorphism, in factor modules, and in Cartesian products of modules (see Proposition 5, Corollary 3, and Theorems 2 and 3). An -module is said to be a multiplication module if each submodule of has the form for some ideal of . In Theorem 1, we determine all -semiprime submodules of finitely generated multiplication modules. Also, we characterize certain semiprime submodules of an arbitrary module in terms of -semiprime submodules (see Theorem 5). Using Theorem 5, we determine all semiprime submodules of modules over quasi-local rings in terms of -semiprime submodules (see Corollary 4). Finally, we characterize reduced modules in terms of -reduced modules (see Theorem 6).

2. Characterization of -Semiprime Submodules

Definition 1. Let be a submodule of with , where is a m.c.s of . is said to be an -semiprime submodule if there exists a fixed , and whenever for some , and then .
Let be a m.c.s of . If we consider the ring as a module over itself, then we say that is an -semiprime ideal if it is an -semiprime submodule of . Note that an ideal of with is an -semiprime ideal if and only if there exists (fixed) , and whenever for some and then .

Proposition 1. Let be a m.c.s of and be an -module. The following statements are satisfied:(i)If is a semiprime submodule of provided that , then is an -semiprime submodule of (ii)If is an -semiprime submodule of and , then is a semiprime submodule of (iii)Every -prime submodule is also an -semiprime submodule

Proof. (i)Let be a semiprime submodule of and . Now, we will show that is an -semiprime submodule of . To see this, take and such that for some . Since is a semiprime submodule, we have , and this implies that for each . Then, is an -semiprime submodule of .(ii)Let be an -semiprime submodule of . Then, we know that , where , and , implies that for a fixed . Since , there is a such that , and so, . Therefore, is a semiprime submodule of .(iii)Suppose that is an -semiprime submodule of . Let for some , and . Since is an -prime submodule, there exists a fixed such that or . If then , and so, the proof is completed. Now, assume that , that is, . This implies that . If we continue in this manner, we conclude that . As is an -prime submodule, we get either or . As and we have , and so, .The converses of Proposition 1 (i) and (iii) need not be true. See the following examples.

Example 1. Consider the -module and the zero submodule . First, note that and . Since , is not a semiprime submodule of . Now, take the m.c.s of , and put . Now, we will show that is an -semiprime submodule. To see this, let for some and . Then, we have . If , then . Otherwise, we have , and so, . Therefore, is an -semiprime submodule.

Example 2. Let and where are distinct prime numbers. Consider the multiplicatively closed subset of . Take the submodule . Then, note that , and also, and for any . Thus, is not an -prime submodule. Also, it is clear that is an -semiprime submodule of
Let be a m.c.s of . Then, is called the quotient ring of . For any m.c.s of the saturation of is defined as [13]. Note that is a m.c.s of containing .

Proposition 2. Let be a m.c.s of and be an -module. The following statements hold:(i)If is a m.c.s of and is an -semiprime submodule of , then is an -semiprime submodule of in case (ii) is an -semiprime submodule of if and only if is an -semiprime submodule of (iii)If is an -semiprime submodule of , then is a semiprime submodule of

Proof. (i)It is straightforward.(ii)Let . It is clear that . Then, we need to show that . Assume that . So, there exists . Since is a unit of , and so, for some . This yields that for some . Now, put . Then, , a contradiction. Thus, . So, by (i), is an -semiprime submodule of . For the converse, let be an -semiprime submodule of . Then, , and thus, . Let for some , and . Since is an -semiprime submodule, there is an such that . As , there exists such that . Then, we conclude that for some . Now, take . Then, we get , and hence, is an -semiprime submodule of .(iii)Suppose that is an -semiprime submodule of . Let for some , and . Then, there exists such that . As is an -semiprime submodule of we get for some . This implies that . Hence, is a semiprime submodule of .The following example shows that the converse of Proposition 2 (iii) is not true in general.

Example 3. Let and , where is the field of rational numbers. Take the submodule and the m.c.s of . It is easy to see that is a vector space over , and thus, is a prime (semiprime) submodule of . Now, we will show that is not -semiprime. Let be an arbitrary element of . Choose a prime number with . Then, note that and . Thus, is not an -semiprime submodule of .

Proposition 3. Let be an -module and be a finite m.c.s of . Suppose that is a submodule of provided that . Then, is an -semiprime submodule of if and only if is a semiprime submodule of .

Proof. Suppose that is an -semiprime submodule of . Then, by Proposition 2 (iii), is a semiprime submodule of . For the converse, take a semiprime submodule of . Let for some , and . Then, we have . Since is a semiprime submodule of we conclude that , and this yields that for some . Now, put . Then, we conclude that , and so, is an -semiprime submodule of .

Lemma 1. Suppose is a submodule of and is a m.c.s of provided that . The following statements are equivalent:(i) is an -semiprime submodule of (ii)There is a fixed and for some implying that for each ideal of and submodule of

Proof. let be an -semiprime submodule of . Suppose that for some ideal of , some submodule of , and . Now, we will show that . Suppose to the contrary. Then, there exist such that . Since and is an -semiprime submodule of we conclude that a contradiction. Therefore, conversely, let for some , and . Now, put and . Then, we have . Hence, by assumption, for a fixed , and so, . Then, is an -semiprime submodule of .
As immediate consequences of the previous lemma, we give the following corollary which will be used in the sequel.

Corollary 1. Suppose that is a m.c.s of and is an ideal of with .The following statements are equivalent:(i) is an -semiprime ideal of (ii)There is a (fixed) for some ideals of and implying that

Proposition 4. Let be an -module and be a m.c.s of . Suppose that is a submodule of with . The following statements hold:(i)If is an -semiprime submodule of , then is an -semiprime ideal of (ii)If is a multiplication module and is an -semiprime ideal of , then is an -semiprime submodule of

Proof. (i)Let for some and . Then, we have for each . Since is an -semiprime submodule, we conclude that , and this yields that . Therefore, is an -semiprime ideal of .(ii)Assume that is a multiplication module and is an -semiprime ideal of . Let for some ideal of submodule of , and . Then, we conclude that . Also, note that, by Corollary 1, there exists a fixed such that . Since is a multiplication module, we have . Then, by Lemma 1, is an -semiprime submodule of .

Corollary 2. Suppose that is a submodule of a multiplication -module and is a m.c.s of such that . Then, the following statements are equivalent:(i) is an -semiprime submodule of (ii)There exists a fixed such that for some submodules of and implying

Proof.  : suppose that is an -semiprime submodule of . Let for some submodules of and . Since is a multiplication module, and for some ideal of . Also, note that . Since is an -semiprime submodule, by Lemma 1, there exists such that , and this yields that . : suppose that for some ideal of , submodule of , and . Then, we have . Now, put , and note that . This implies that . Then, by assumption, there exists a fixed such that . Then, by Lemma 1, is an -semiprime submodule of .

Theorem 1. Let be a submodule of a finitely generated multiplication -module and be a m.c.s of with . The following statements are equivalent:(i) is an -semiprime submodule of (ii) is an -semiprime ideal of (iii) for some -semiprime ideal of with

Proof.  : follows from Proposition 4 (i). : it is straightforward. : suppose that for some -semiprime ideal of with . Assume that for some ideal of , some submodule of , and . Then, we obtain . As is a finitely generated multiplication module, by [14], Theorem 9 Corollary, we have . Since is an -semiprime ideal of , by Corollary 1, for a fixed such that , this yields that . Then, by Lemma 1, is an -semiprime submodule of .

Proposition 5. Let be an -homomorphism. The following statements are satisfied:(i)If is an -semiprime submodule of such that , then is an -semiprime submodule of (ii)If is an epimorphism and is an -semiprime submodule of such that , then is an -semiprime submodule of

Proof. (i)Let for some , and . Then, we conclude that . Since is an -semiprime submodule, we have for some . Then, we get , and thus, is an -semiprime submodule of .(ii)Let for some , and . As is an epimorphism, we can write for some , and so, Since we conclude that . Since is an -semiprime submodule, there exists such that , and so, we obtain . Consequently, is an -semiprime submodule of .

Corollary 3. Suppose that is a m.c.s of and is a submodule of . Then, the following statements are satisfied:(i)If is an -semiprime submodule of with , then is an -semiprime submodule of .(ii)Suppose that is a submodule of with . Then, is an -semiprime submodule of if and only if is an -semiprime submodule of .

Proof. (i)Consider the injection defined by for all . Then, note that . Now, we will show that . Assume that . Then, we have , and thus, , a contradiction. By Proposition 5 (i), we can say that is an -semiprime submodule of .(ii)Assume that is an -semiprime submodule of . Consider the natural epimorphism , defined by , for all . By Proposition 5 (ii), is an -semiprime submodule of . For the converse, let be an -semiprime submodule of . Take and with for some . Then, we get . Since is an -semiprime submodule of , for a fixed , we conclude that . This implies that , and hence, is an -semiprime submodule of .Let be an module and be a m.c.s of for each , where . Suppose that and . Then, is an -module with componentwise addition and scalar multiplication, and note that is a m.c.s of . Also, each submodule of has the form where is a submodule of for each .

Theorem 2. Suppose that is an -module, is a submodule of , and is a m.c.s of for each . Let and . The following statements are equivalent for :(i) is an -semiprime submodule of .(ii) is an -semiprime submodule of and or , and is an -semiprime submodule of or is an -semiprime submodule of for each .

Proof.  : let be an -semiprime submodule of . Then, we have , and this yields that or . Without loss of generality, we may assume that and . We must show that is an -semiprime submodule of . To prove this, take such that for some . Then, . Since is an -semiprime submodule of , there exists a fixed such that . This implies that . Hence, is an -semiprime submodule of . One can similarly show that if then is an -semiprime submodule of . Also, if then a similar argument shows that is an -semiprime submodule of for each . : let and be an -semiprime submodule of . Then, we have . Let for some , where . This implies that , and so, for a fixed since is an -semiprime submodule of . Now, take . Then, . Similarly, we can show that is an -semiprime submodule of in other cases.

Theorem 3. Let be an module and be a m.c.s of for each . Suppose that , and . Let where is a submodule of for each . The following statements are equivalent:(i) is an -semiprime submodule of (ii) is an -semiprime submodule of for each , and for each

Proof. We use mathematical induction to prove the claim . For , the result is clear. If , the claim follows from Theorem 2. Suppose that and are equivalent for each . Now, we will show that the claim is true for . Let and also and . Note that and also . Then, by Theorem 2, is an -semiprime submodule of if and only if is an -semiprime submodule of and or and is an -semiprime submodule of or is an -semiprime submodule of , and is an -semiprime submodule of . The rest follows from induction hypothesis.

Theorem 4. Let be a submodule of and be a m.c.s of such that . Then, is an -semiprime submodule of if and only if is a semiprime submodule of for some .

Proof. Let be an -semiprime submodule of , and put . Now, we will show that is a semiprime submodule of . Let for some and . Then, we get . If , then we have . So, assume that . Then, clearly we have , and this gives . Then, we conclude that . Hence, is a semiprime submodule of . Conversely, assume that is a semiprime submodule of for some . Let for some , and . Since is a semiprime submodule and we conclude that , and hence, . Therefore, is an -semiprime submodule of .

Theorem 5. Let be a submodule of such that where is the Jacobson radical of . The following statements are equivalent:(i) is a semiprime submodule of (ii) is an -semiprime submodule of for each maximal ideal m of R

Proof.  : suppose that is a semiprime submodule of . Then, by Proposition 1, is an -semiprime submodule of for each maximal ideal m of R.  let for some , and . As is an -semiprime submodule of there exists an such that . Now, consider the set . Now, we will show that . Assume that . Then, there exists a maximal ideal of containing . By the definition of , there exists such that . Since we have which is a contradiction. Thus, , and so, there exists such that for some . Since for each we conclude that . Therefore, is an -semiprime submodule of .As immediate consequence of the previous theorem, we give the following result.

Corollary 4. Let be a module over a quasi-local ring . Suppose that is a submodule of . The following statements are equivalent:(i) is a semiprime submodule of (ii) is an -semiprime submodule of

Definition 2. Let be an -module and be a m.c.s of is said to be an -reduced module if there exists , and whenever , where , and , then .

Proposition 6. Suppose that is an -module and is a m.c.s of . The following statements are satisfied:(i)If is a reduced module, then is an -reduced module. In particular, the converse holds if where .(ii)If is an -torsion-free module, then is an -reduced module.(iii) is an -reduced module if and only if the zero submodule is an -semiprime submodule.(iv)Let be a submodule of with . Then, is an -semiprime submodule if and only if -module is an -reduced module.(v)If is an -reduced module, then is a reduced module.

Proof. (i)The claim “reduced module implies the -reduced module” is obvious. Let be an -reduced module such that . Let for some , , and . Since is an -reduced module, there exists such that As , we have , and so, . Hence, is a reduced module.(ii)Let be an -torsion-free module and for some , and . Since is an -torsion-free module, there exists such that or . If , then which completes the proof. So, assume that . Since , we conclude that since is an -torsion-free module. If we continue in the previous way, we conclude that . Since is an -torsion-free module, we get either or . If then , which is a contradiction so that , and this yields .(iii)It follows from Definitions 1 and 2.(iv)It follows from (iii).(v)Let be an -reduced module. Then by (iii), 0 is an -semiprime submodule of . Again by Proposition 2 (iii), is a semiprime submodule of . Thus, is a reduced module.Now, we will characterize reduced modules in terms of -reduced modules.

Theorem 6. The following statements are equivalent for any -module :(i) is a reduced module(ii) is an -reduced module for each (iii) is an -reduced module for each

Proof.  : it follows from Proposition 6. : it follows from the fact that . : let be an -reduced module for each . Choose and such that . Since is an -reduced module for each , there exists such that . Now, consider the set and . Note that is not empty since is an -reduced module. Similar argument in Theorem 5 shows that , and so, there exists such that and . This yields that . Hence, is a reduced module.

3. Conclusion

This paper is mainly concerned with S-semiprime submodules of modules over commutative rings. We first investigate some properties of S-semiprime submodules similar to semiprime submodules. Then, we introduce S-reduced modules and give some new characterizations of semiprime submodules and reduced modules in terms of these concepts.

Data Availability

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Conflicts of Interest

The authors declare no conflicts of interest.