Abstract

Let be a commutative ring with an identity. The purpose of this paper is to introduce and investigate the notion of strong -submodules of an -module as an extension of -submodules and quasi -submodules.

1. Introduction

Throughout this paper, will denote a commutative ring with identity and will denote the ring of integers.

For an ideal of , let be the set of maximal ideals of containing and be the set of all maximal ideals of . A proper ideal of is called a -ideal if whenever any two elements of are contained in the same set of maximal ideals and contains one of them, it also contains the other one [1]. Intersections of maximal ideals are -ideals and they are called strong -ideals. It is easy to check that a proper ideal of is a strong -ideal if and only if for ideals of with and , then . In general, the strong -ideals are not the only -ideals [[1], p. 281].

For each , let , be the intersection of all minimal prime ideals containing . A proper ideal of is called a -ideal if for each , we have [2]. -ideals have been studied in [3] under the name of -ideals.

Let be an -module. A proper submodule of is said to be prime if for any and with , we have or [4, 5]. The intersection of all prime submodules of containing a submodule of is said to be the prime radical of and denoted by . In case does not contain in any prime submodule, the prime radical of is defined to be [6]. A prime submodule is a minimal prime submodule over if is a minimal element of the set of all prime submodules of that contain [7]. A minimal prime submodule of means a minimal prime submodule over the 0 submodules of . The set of all minimal prime submodules of will be denoted by . The intersection of all minimal prime submodules of containing a submodule of is denoted by . In case does not contain in any minimal prime submodule of , is defined to be . Also, the intersection of all minimal prime submodules of containing is denoted by . In case does not contain in any minimal prime submodule of , is defined to be . If is a submodule of , define .

An -module is said to be a multiplication module if for every submodule of , there exists an ideal of such that [8]. An -module is said to be reduced if the intersection of all the prime submodules of is equal to zero [9].

In [10, 11], the notions of -submodules and quasi -submodules of an -module as an extension of -ideals were introduced and some of their properties were investigated when is a reduced multiplication -module. A proper submodule of an -module is said to be a -submodule of if for all [10]. A proper submodule of an -module is said to be a quasi -submodule of if for all [11]. In this paper, we define the notion of strong -submodules of an -module as a generalization of -submodules and quasi -submodules. We say that a proper submodule of an -module is a strong -submodule of if for all submodules of . Among other results, we give some characterizations for strong -submodules of an -module , in particular, when is a Noetherian reduced multiplication module.

2. Main Results

We begin Section 2 by introducing the concept of the strong -submodule of a module.

Definition 1. We say that a proper submodule of an -module is a strong -submodule of if for all submodules of . Also, we say that a proper ideal of is a strong -ideal if is a strong -submodule of an -module .

Remark 1. Let be an -module. Clearly, if is a strong -submodule of , then is a -submodule of . The converse holds when every submodule of is a cyclic -module. However, as we see (Example 4.2 in [12]), the converse is not true in general.

Remark 2. If is a strong -submodule of , then for every submodule of , we have . Clearly, every minimal prime submodule of is a strong -submodule of . Also, the family of strong -submodules of is closed under the intersection. Therefore, if , then is a strong -submodule of and it is contained in every strong -submodule of .

Proposition 1. Let be a submodule of an -module . Then, is a strong -submodule of if and only if is an intersection of minimal prime submodules of .

Proof. Since , we have ; therefore, . The converse is clear.

Lemma 1. Let be an -module. A submodule of is a strong -submodule if and only if , where is the collection of all submodules of .

Proof. This is straightforward.
A submodule of an -module is said to be a multiple of , provided that for some . If every submodule of is a multiple of , then is said to be a principal ideal multiplication module [13].

Remark 3. Clearly, if is a strong -submodule of , then is a quasi -submodule of . The converse holds when is a principal ideal multiplication module.

Proposition 2. Let be a proper submodule of an -module . Then, as an -submodule is a strong -submodule if and only if as an -submodule is a strong -submodule.

Proof. This is straightforward.

Remark 4. Let be a faithful multiplication -module. Then, for each ideal of . The reverse inclusion holds when is a finitely generated -module (see Theorem 2.8 in [11]).

Theorem 1. Let be a strong -submodule of a faithful multiplication -module . Then, is a strong -ideal of . The converse holds when is a finitely generated -module.

Proof. Let be an ideal of such that . Then, and so by assumption, . Hence, by Remark 4. It follows that and so is a strong -ideal of . For the converse, let be a finitely generated -module and be a submodule of such that . Thus, . Now, by assumption, . By Remark 4, . Therefore, , as needed.

Corollary 1. Let be a finitely generated faithful multiplication -module. If is a strong -ideal of , then is a strong -submodule of .

Proof. By Theorem 10 in [14], . Now, the result follows from Theorem 1.

Theorem 2. Let be a proper submodule of an -module . Then, the following are equivalent:(a) is a strong -submodule of (b)For submodules ,of, and imply that (c)For submodules ,of, and imply that (d)If is a submodule of , a submodule of , and , then

Proof.   Let , be submodules of such that and . By part (a), . Thus,   Let , be submodules of such that and . Then, . Thus, by part (b),   Let be a submodule of . One can see that . Now, by part (c),   Let be a submodule of , a submodule of , and . Then, . Hence, by part (a),   This is clear.

Lemma 2. Let be a reduced multiplication -module and be a minimal prime submodule of . If is a finitely generated submodule of such that , then .

Proof. This follows from the proof of (Theorem 3.6 in [9]).

Proposition 3. Let be a reduced multiplication -module. Then, the following are equivalent:(a)For submodules of , and imply that (b)For submodules of , and imply that

Proof.   Let be submodules of . As is a multiplication -module, there exist ideals and of such that and . Assume that and . Then, . Hence, by Theorem 2.5 in [11]. This implies that . Now, as , we have by part (a).  This is clear.

Theorem 3. Let be a reduced multiplication -module. Then, for each finitely generated submodule of , we have the following:(a)(b)

Proof. (a)If , then by Lemma 2. Thus, . Therefore, . On the other hand, if , then there exists . Hence, for any , we have . Since and is prime, . This implies that . Thus, .(b)Clearly, . Now, let be a minimal prime submodule of containing . Then, there exists by Lemma 2. Thus, for any , we have . It follows that . Therefore, . Thus, .

Corollary 2. Let be a reduced multiplication -module. Then, for each finitely generated submodule of , .

Proof. This follows from Theorem 2 in [10] and Theorem 3(b).

Theorem 4. Let be a submodule of a reduced multiplication -module . Then, for finitely generated submodules and of , the following are equivalent:(a) and imply that (b) and imply that (c) and imply that

Proof.   Assume that and . Then, . Thus, by part (a), .  Assume that and . Then, . So, by Theorem 3(b), . Hence, by part (b).  Assume that and . Then, by Corollary 2. Thus, So, by part (c).

Proposition 4. Let be a strong -submodule of a faithful multiplication -module . Then, for each , is a strong -submodule of .

Proof. As is a faithful multiplication -module, one can see that for each submodule of . Suppose that and . Then, . So, by assumption, . Since , we have . Thus, .

Theorem 5. Let be a Noetherian reduced multiplication -module. Then, the following are equivalent:(a)For submodules and of , and imply that (b)For a submodule of , implies that (c)For submodules and of , and imply that

Proof.   Let be a submodule of . As is Noetherian, is finitely generated. Thus, by Theorem 3. Hence, by part (b), .  Let , be submodules of such that and . Then, . By part (c), . Thus, .  This follows from Theorem 4.Now, we have the following corollary.

Corollary 3. Let be a Noetherian reduced multiplication -module. Then, for a proper submodule of , the following are equivalent:(a) is a strong -submodule of (b) is an intersection of minimal prime submodules of (c) is a strong -ideal of (d)For a submodule of , implies that (e)For submodules and of , and imply that (f)For submodules and of , and imply that (g)For submodules , of , and imply that (h)If is a submodule of , a submodule of , and , then ;(i), where is the collection of all submodules of .An -module is said to be a comultiplication module if for every submodule of there exists an ideal of such that equivalently for each submodule of , we have [15].

Example 1. Let be a Noetherian reduced multiplication and comultiplication -module. Then, every proper submodule of is a strong -submodule of . In particular,(a)If is a prime number and , then every proper submodule of the -module is a strong -submodule(b)If is square-free, every proper submodule of the -module is a strong -submodule

Theorem 6. Let be a Noetherian reduced multiplication -module and be a strong -submodule of . Then, every minimal prime submodule over is a prime strong -submodule of . In particular, is a strong -submodule of .

Proof. Let be a minimal prime submodule over . Assume that , where are submodules of with . Since is a minimal prime submodule of , by Lemma 2, there exists . Thus, and . Clearly, . As is a strong -submodule of , we have . As and is a prime submodule, as needed. The last assertion is clear.

Corollary 4. Let be a Noetherian reduced multiplication -module. If is the natural epimorphism, where is a strong -submodule of , then every strong -submodule of contracts to a strong -submodule of .

Proposition 5. Let be a Noetherian reduced multiplication -module. Let for be a proper submodule of such that for each , and are co-prime ideals of . Then, is a strong -submodule of if and only if each for is a strong -submodule of .

Proof. Assume that is a strong -submodule of and . We show that is a strong -submodule of . So, assume that for some submodules of with . Since, for each , and are co-prime ideals of , and are co-prime ideals of . Thus, for some and . So, and . Now, we have . Thus, is a strong -submodule of which implies that . Now, since , we have and we are done. The converse is clear.

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Conflicts of Interest

The author declares that there are no conflicts of interest.