1. Introduction

Let be a commutative ring with identity, and let be a unital -module. A commutative ring is called a quasilocal ring if it has a unique maximal ideal and denoted by . Let be a submodule of and the ideal denoted by . Let be an ideal of and the radical of denoted by and defined . A proper submodule of is said to be prime (weakly prime) if (), then either or (either or ), where and . A proper submodule of is said to be primary (weakly primary) if (), then either or for some positive integer (either or for some positive integer ), where and . It is clear that every primary submodule is weakly primary. However, since 0 is always weakly primary (by definition), so a weakly primary submodule need not be primary. A proper submodule of an -module said to be maximal if there is no submodule of such that . A submodule of is called -submodule of , provided that contained in a finite union of submodules must be contained one of those submodules. is called -module if every submodule of is a -submodule. A submodule of is called -submodule of , provided that contained in a finite union of primary submodules that must be contained in one of those primary submodules. is called -module if every submodule of is a -submodule. A submodule of is called -submodule of , provided that contained in a finite union of maximal submodules that must be contained in one of those submodules. is called -module if every maximal submodule of is a -submodule. An -module is called a multiplication module, provided that for each submodule of , there exists an ideal of such that . If is a ring and an -module, the subset of is defined by . Obviously, if is an integral domain, then is a submodule of . In this paper, we investigate finite unions of weakly primary submodules of -modules.

2. On Weakly Primary Submodules

It is clear that every primary submodule is a weakly primary submodule. However, since 0 is always weakly primary (by definition), a weakly primary submodule need not be primary, but we have the following results.

Proposition 2.1. Let be an -module with . Then, every weakly primary submodule of is primary.

Proof. Let be a weakly primary submodule of . Suppose that , where , . If , weakly primary gives or for some positive integer . If , then or , since . So, is primary.

Proposition 2.2. Let be a module over a quasilocal ring (,) with . Then, every proper submodule of is weakly primary.

Proof. Let be a proper submodule of and , where and . If is a unite, then . Let is not a unite, so , a contradiction. Hence, is weakly primary.

Lemma 2.3. Let be an -module. Assume that and are submodules of such that with . Then, the following hold:(i)if is a weakly primary submodule of , then is a weakly primary submodule of , (ii)if and are weakly primary submodules, then is weakly primary.

Proof. (i) Let , where and . If , then , which is a contradiction. If , weakly primary gives either or for some positive integer , hence either or (since we have ), as required.
(ii) Let , where and , so . If , then weakly primary gives either or . So, we may assume that . Then, . Since is weakly primary, we get either or for some positive integer . Thus, or for some positive integer , as required.

Theorem 2.4. Let be a secondary -module and a nonzero weakly primary -submodule of . Then, is secondary.

Proof. Let . If for some . Then, , so is nilpotent on . Suppose that ; we show that divides . Assume that . So, for some . We may assume that . Hence, and for any positive integer (since ), then weakly primary gives . Thus, , as needed.

Theorem 2.5. Let be an -module, a secondary -submodule of , and a weakly primary submodule of . Then, is secondary.

Proof. The proof is straightforward.

Proposition 2.6. Let be a module over a commutative ring and a multiplicatively closed subset of . Let be a weakly primary submodule of such that . Then, is a weakly primary submodule of -module .

Proof. Let , where and . So, for some and , hence there exists such that (because if , , a contradiction) and , so weakly primary gives . Hence, or , thus or , as needed.

Lemma 2.7. Let be a module over a quasilocal ring and a weakly primary submodule of . Then, .

Proof. Let and . We show that . We may assume that . We have , so for some and . There exists such that . If , then , a contradiction. So, and , then . Thus, . Clearly, , so the proof is complete.

Theorem 2.8. Let be a module over a quasilocal ring . Then, there exists a one to one correspondence between the weakly primary submodules of and the weakly primary submodules of -module .

Proof. Let be a weakly primary submodule of . So, for some submodule of . We show that is weakly primary submodule of . Let , so (if , then for some , , a contradiction). Hence, for some positive integer by Lemma 2.7 or , since is weakly primary. Thus, for some positive integer or , as required. Let be a weakly primary submodule of . Then, by Proposition 2.6, is weakly primary submodule of .

3. Unions of Weakly Primary Submodules

Definition 3.1. Let be a module over a commutative ring and a submodule of ; is called a -submodule of , provided that contained in a finite weakly primary submodules of must be contained in one of those weakly primary submodules. is called a -module if every submodule of is a -submodule.

Clearly, every -module is -module, and every -module is -module and also, every -module is -module.

Theorem 3.2. Let be a module over a commutative ring and a submodule of . Then,(i)if is a -submodule of and is a weakly primary submodule of such that , then is a -submodule of -module ,(ii)if is a -submodule of , then is a -submodule of .

Proof. (i) Let , where ’s are weakly primary submodules of . Then, by Lemma 2.3, there exists weakly primary submodules of such that for . So, , so . Hence, for some , since is -submodule. Thus , for some , as needed.
(ii) Let , where ’s are weakly primary submodule of . So, . Therefore, ’s are weakly primary submodules of , so for some . Hence, for some , as required.

Theorem 3.3. Let be an -module, a weakly primary submodule of , and a -module. Then, is a -module.

Proof. By Theorem 3.2.

Theorem 3.4. Let be a module over quasi local ring . Then, is a -module if and only if module is a -module.

Proof. Let be a -module, and let be a submodule of -module such that , where ’s are weakly primary submodules of . So, for some submodule of and for some weakly primary submodules of by Theorem 2.8. Hence, , so , thus by [1, Theorem  2.8]. Therefore, by hypothesis for some . So for some as needed.
Conversely, let -module is a -module, and let be a submodule of such that , where ’s are weakly primary submodules of . So, . Thus, for some by hypothesis. Then, for some . So is a -module.

Theorem 3.5. Let be a finitely generated -module. Then, is a -module if and only if every submodule in such that , where ’s are weakly primary submodules implies that for some .

Proof. Let be a finitely generated -module. Suppose that be a submodule of such that , where ’s are weakly primary submodules of . For each , is a maximal submodule containing . Then, , and so, for some by hypothesis. Since , we have .
Conversely, let be a submodule of such that , where ’s are maximal submodules of . Since every maximal submodule is weakly primary submodule, then for some by hypothesis. Therefore, since , then , so for some . The proof is complete.

Definition 3.6. By a chain of weakly primary submodules of an -module , we mean a finite strictly increasing sequence ; the weakly primary dimension of this chain is . We define the weakly primary dimension of to be the supremum of the lengths of all chains of weakly primary submodules in .

Theorem 3.7. Let be a finitely generated -module with weakly primary dimension 1. Then, is a -module if and only if is a -module.

Proof. Let be a -module. Since every -module is a -module, so is a -module.
Conversely, let be a -module. Let be a nonzero submodule of such that , where ’s are weakly primary submodules of . We may assume that for all . By Theorem 3.5, for some . There exists a maximal submodule of such that . Since 0 is a weakly primary submodule of , so we have . Hence, since weakly primary dimension of is 1; . Consequently, for some , as needed.

Theorem 3.8. Let be an -module with . Then, is a -module if and only if is a -module.

Proof. Let be a -module. Then, is a -module since every primary submodule is weakly primary. Let be a -module and a submodule of such that where ’s are weakly primary submodules of . By Proposition 2.1, ’s are primary submodule, so for some , as needed.