`ISRN Discrete MathematicsVolumeΒ 2011Β (2011), Article IDΒ 939687, 5 pageshttp://dx.doi.org/10.5402/2011/939687`
Research Article

## Notes on the Union of Weakly Primary Submodules

Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

Received 30 August 2011; Accepted 18 October 2011

Copyright Β© 2011 Peyman Ghiasvand and Farkhonde Farzalipour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be a commutative ring with identity, and let be an -module. A proper submodule of is said to be weakly primary if for and , which implies that either or for some positive integer . In this paper, we study weakly primary submodules, and we investigate the union of weakly primary submodules of -modules.

#### 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.

#### References

1. F. Farzalipour and P. Ghiasvand, βQuasi multiplication modules,β Thai Journal of Mathematics, vol. 2, pp. 361β366, 2009.