About this Journal Submit a Manuscript Table of Contents
ISRN Discrete Mathematics
VolumeΒ 2011Β (2011), Article IDΒ 939687, 5 pages
http://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

Academic Editor: U. A.Β Rozikov

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 𝑅 be a commutative ring with identity, and let 𝑀 be an 𝑅 -module. A proper submodule 𝑁 of 𝑀 is said to be weakly primary if 0 β‰  π‘Ÿ π‘š ∈ 𝑁 for π‘Ÿ ∈ 𝑅 and π‘š ∈ 𝑀 , which implies that either π‘š ∈ 𝑁 or π‘Ÿ 𝑛 𝑀 βŠ† 𝑁 for some positive integer 𝑛 . In this paper, we study weakly primary submodules, and we investigate the union of weakly primary submodules of 𝑅 -modules.

1. Introduction

Let 𝑅 be a commutative ring with identity, and let 𝑀 be a unital 𝑅 -module. A commutative ring 𝑅 is called a quasilocal ring if it has a unique maximal ideal 𝑃 and denoted by ( 𝑅 , 𝑃 ) . Let 𝑁 be a submodule of 𝑀 and the ideal { π‘Ÿ ∈ 𝑅 ∢ π‘Ÿ 𝑀 βŠ† 𝑁 } denoted by ( 𝑁 ∢ 𝑀 ) . Let 𝐼 be an ideal of 𝑅 and the radical of 𝐼 denoted by √ 𝐼 and defined √ 𝐼 = { π‘Ÿ ∈ 𝑅 ∢ π‘Ÿ 𝑛 ∈ 𝐼 f o r s o m e p o s i t i v e i n t e g e r 𝑛 } . A proper submodule 𝑁 of 𝑀 is said to be prime (weakly prime) if π‘Ÿ π‘š ∈ 𝑁 ( 0 β‰  π‘Ÿ π‘š ∈ 𝑁 ), then either π‘š ∈ 𝑁 or π‘Ÿ 𝑀 βŠ† 𝑁 (either π‘š ∈ 𝑁 or π‘Ÿ 𝑀 βŠ† 𝑁 ), where π‘Ÿ ∈ 𝑅 and π‘š ∈ 𝑀 . A proper submodule 𝑁 of 𝑀 is said to be primary (weakly primary) if π‘Ÿ π‘š ∈ 𝑁 ( 0 β‰  π‘Ÿ π‘š ∈ 𝑁 ), then either π‘š ∈ 𝑁 or π‘Ÿ 𝑛 𝑀 βŠ† 𝑁 for some positive integer 𝑛 (either π‘š ∈ 𝑁 or π‘Ÿ 𝑛 𝑀 βŠ† 𝑁 for some positive integer 𝑛 ), where π‘Ÿ ∈ 𝑅 and π‘š ∈ 𝑀 . 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 𝑁 of an 𝑅 -module 𝑀 said to be maximal if there is no submodule 𝐾 of 𝑀 such that 𝑁 ΜΈ ΜΈ βŠ† 𝐾 βŠ† 𝑀 . A submodule 𝑁 of 𝑀 is called 𝑒 -submodule of 𝑀 , provided that 𝑁 contained in a finite union of submodules must be contained one of those submodules. 𝑀 is called 𝑒 -module if every submodule of 𝑀 is a 𝑒 -submodule. A submodule 𝑁 of 𝑀 is called 𝑒 𝑝 π‘Ÿ -submodule of 𝑀 , provided that 𝑁 contained in a finite union of primary submodules that must be contained in one of those primary submodules. 𝑀 is called 𝑒 𝑝 π‘Ÿ -module if every submodule of 𝑀 is a 𝑒 𝑝 π‘Ÿ -submodule. A submodule 𝑁 of 𝑀 is called 𝑒 π‘š -submodule of 𝑀 , provided that 𝑁 contained in a finite union of maximal submodules that must be contained in one of those submodules. 𝑀 is called 𝑒 π‘š -module if every maximal submodule of 𝑀 is a 𝑒 π‘š -submodule. An 𝑅 -module 𝑀 is called a multiplication module, provided that for each submodule 𝑁 of 𝑀 , there exists an ideal 𝐼 of 𝑅 such that 𝑁 = 𝐼 𝑀 . If 𝑅 is a ring and 𝑀 an 𝑅 -module, the subset 𝑇 ( 𝑀 ) of 𝑀 is defined by 𝑇 ( 𝑀 ) = { π‘š ∈ 𝑀 ∢ π‘Ÿ π‘š = 0 f o r s o m e 0 β‰  π‘Ÿ ∈ 𝑅 } . Obviously, if 𝑅 is an integral domain, then 𝑇 ( 𝑀 ) is a submodule of 𝑀 . In this paper, we investigate finite unions of weakly primary submodules of 𝑅 -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 𝑀 be an 𝑅 -module with 𝑇 ( 𝑀 ) = 0 . Then, every weakly primary submodule of 𝑀 is primary.

Proof. Let 𝑁 be a weakly primary submodule of 𝑀 . Suppose that π‘Ÿ π‘š ∈ 𝑁 , where π‘Ÿ ∈ 𝑅 , π‘š ∈ 𝑀 . If 0 β‰  π‘Ÿ π‘š ∈ 𝑁 , 𝑁 weakly primary gives π‘š ∈ 𝑁 or π‘Ÿ 𝑛 𝑀 βŠ† 𝑁 for some positive integer 𝑛 . If π‘Ÿ π‘š = 0 , then π‘Ÿ = 0 or π‘š = 0 , since 𝑇 ( 𝑀 ) = 0 . So, 𝑁 is primary.

Proposition 2.2. Let 𝑀 be a module over a quasilocal ring ( 𝑅 , 𝑃 ) with 𝑃 𝑀 = 0 . Then, every proper submodule of 𝑀 is weakly primary.

Proof. Let 𝑁 be a proper submodule of 𝑀 and 0 β‰  π‘Ÿ π‘š ∈ 𝑁 , where π‘Ÿ ∈ 𝑅 and π‘š ∈ 𝑀 . If π‘Ÿ is a unite, then π‘š ∈ 𝑁 . Let π‘Ÿ is not a unite, so π‘Ÿ π‘š ∈ 𝑃 𝑀 = 0 , a contradiction. Hence, 𝑁 is weakly primary.

Lemma 2.3. Let 𝑀 be an 𝑅 -module. Assume that 𝑁 and 𝐾 are submodules of 𝑀 such that 𝐾 βŠ† 𝑁 with 𝑁 β‰  𝑀 . Then, the following hold:(i)if 𝑁 is a weakly primary submodule of 𝑀 , then 𝑁 / 𝐾 is a weakly primary submodule of 𝑀 / 𝐾 , (ii)if 𝐾 and 𝑁 / 𝐾 are weakly primary submodules, then 𝑁 is weakly primary.

Proof. (i) Let 0 β‰  π‘Ÿ ( π‘š + 𝐾 ) = π‘Ÿ π‘š + 𝐾 ∈ 𝑁 / 𝐾 , where π‘Ÿ ∈ 𝑅 and π‘š ∈ 𝑀 . If π‘Ÿ π‘š = 0 , then π‘Ÿ ( π‘š + 𝐾 ) = 0 , which is a contradiction. If π‘Ÿ π‘š β‰  0 , 𝑁 weakly primary gives either π‘š ∈ 𝑁 or π‘Ÿ 𝑛 ∈ ( 𝑁 ∢ 𝑅 𝑀 ) for some positive integer 𝑛 , hence either π‘š + 𝐾 ∈ 𝑁 / 𝐾 or π‘Ÿ 𝑛 ∈ ( 𝑁 / 𝐾 ∢ 𝑅 𝑀 / 𝐾 ) (since we have ( 𝑁 ∢ 𝑅 𝑀 ) = ( 𝑁 / 𝐾 ∢ 𝑅 𝑀 / 𝐾 ) ), as required.
(ii) Let 0 β‰  π‘Ÿ π‘š ∈ 𝑁 , where π‘Ÿ ∈ 𝑅 and π‘š ∈ 𝑀 , so π‘Ÿ ( π‘š + 𝐾 ) = π‘Ÿ π‘š + 𝐾 ∈ 𝑁 / 𝐾 . If π‘Ÿ π‘š ∈ 𝐾 , then 𝐾 weakly primary gives either π‘š ∈ 𝐾 βŠ† 𝑁 or π‘Ÿ 𝑛 ∈ ( 𝐾 ∢ 𝑅 𝑀 ) βŠ† ( 𝑁 ∢ 𝑅 𝑀 ) . So, we may assume that π‘Ÿ π‘š βˆ‰ 𝐾 . Then, 0 β‰  π‘Ÿ ( π‘š + 𝐾 ) ∈ 𝑁 / 𝐾 . Since 𝑁 / 𝐾 is weakly primary, we get either π‘š + 𝑁 ∈ 𝑁 / 𝐾 or π‘Ÿ 𝑛 ∈ ( 𝑁 / 𝐾 ∢ 𝑅 𝑀 / 𝐾 ) = ( 𝑁 ∢ 𝑀 ) for some positive integer 𝑛 . Thus, π‘š ∈ 𝑁 or π‘Ÿ 𝑛 ∈ ( 𝑁 ∢ 𝑅 𝑀 ) for some positive integer 𝑛 , as required.

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

Proof. Let π‘Ÿ ∈ 𝑅 . If π‘Ÿ 𝑛 𝑀 = 0 for some 𝑛 ∈ 𝑁 . Then, π‘Ÿ 𝑛 𝑁 βŠ† π‘Ÿ 𝑛 𝑀 = 0 , so π‘Ÿ is nilpotent on 𝑁 . Suppose that π‘Ÿ 𝑀 = 𝑀 ; we show that π‘Ÿ divides 𝑁 . Assume that 𝑛 ∈ 𝑁 . So, 𝑛 = π‘Ÿ π‘š for some π‘š ∈ 𝑀 . We may assume that 0 β‰  π‘Ÿ π‘š . Hence, 0 β‰  π‘Ÿ π‘š ∈ 𝑁 and π‘Ÿ 𝑛 𝑀 ΜΈ βŠ† 𝑁 for any positive integer 𝑛 (since π‘Ÿ 𝑀 = 𝑀 ), then 𝑁 weakly primary gives π‘š ∈ 𝑁 . Thus, π‘Ÿ 𝑁 = 𝑁 , as needed.

Theorem 2.5. Let 𝑀 be an 𝑅 -module, 𝑁 a secondary 𝑅 -submodule of 𝑀 , and 𝐾 a weakly primary submodule of 𝑀 . Then, 𝑁 ∩ 𝐾 is secondary.

Proof. The proof is straightforward.

Proposition 2.6. Let 𝑀 be a module over a commutative ring 𝑅 and 𝑆 a multiplicatively closed subset of 𝑅 . Let 𝑁 be a weakly primary submodule of 𝑀 such that √ ( 𝑁 ∢ 𝑀 ) ∩ 𝑆 = βˆ… . Then, 𝑆 βˆ’ 1 𝑁 is a weakly primary submodule of 𝑆 βˆ’ 1 𝑅 -module 𝑆 βˆ’ 1 𝑀 .

Proof. Let 0 / 1 β‰  π‘Ÿ / 𝑠 β‹… π‘š / 𝑑 ∈ 𝑆 βˆ’ 1 𝑁 , where π‘Ÿ / 𝑠 ∈ 𝑆 βˆ’ 1 𝑅 and π‘š / 𝑑 ∈ 𝑆 βˆ’ 1 𝑀 . So, 0 / 1 β‰  π‘Ÿ π‘š / 𝑠 𝑑 = 𝑛 / 𝑑 β€² for some 𝑛 ∈ 𝑁 and 𝑑 β€² ∈ 𝑆 , hence there exists 𝑠 β€² ∈ 𝑆 such that 0 β‰  𝑠 β€² 𝑑 β€² π‘Ÿ π‘š = 𝑠 β€² 𝑠 𝑑 𝑛 ∈ 𝑁 (because if 𝑠 β€² 𝑑 β€² π‘Ÿ π‘š = 0 , π‘Ÿ π‘š / 𝑠 𝑑 = 𝑠 β€² 𝑑 β€² π‘Ÿ π‘š / 𝑠 β€² 𝑑 β€² 𝑠 𝑑 = 0 / 1 , a contradiction) and √ 𝑠 β€² 𝑑 β€² βˆ‰ ( 𝑁 ∢ 𝑀 ) , so 𝑁 weakly primary gives 0 β‰  π‘Ÿ π‘š ∈ 𝑁 . Hence, π‘š ∈ 𝑁 or π‘Ÿ 𝑛 ∈ ( 𝑁 ∢ 𝑀 ) , thus ( π‘Ÿ / 𝑠 ) 𝑛 ∈ 𝑆 βˆ’ 1 ( 𝑁 ∢ 𝑅 𝑀 ) βŠ† ( 𝑆 βˆ’ 1 𝑁 ∢ 𝑆 βˆ’ 1 𝑅 𝑆 βˆ’ 1 𝑀 ) or π‘š / 𝑑 ∈ 𝑆 βˆ’ 1 𝑁 , as needed.

Lemma 2.7. Let 𝑀 be a module over a quasilocal ring ( 𝑅 , 𝑃 ) and 𝑁 a weakly primary submodule of 𝑀 . Then, ( 𝑁 ∢ 𝑅 𝑀 ) 𝑃 = ( 𝑁 𝑃 ∢ 𝑅 𝑃 𝑀 𝑃 ) .

Proof. Let π‘Ÿ / 𝑠 ∈ ( 𝑁 𝑃 ∢ 𝑅 𝑃 𝑀 𝑃 ) and π‘š ∈ 𝑀 . We show that π‘Ÿ π‘š ∈ 𝑁 . We may assume that π‘Ÿ π‘š β‰  0 . We have π‘Ÿ / 𝑠 β‹… π‘š / 1 ∈ 𝑆 βˆ’ 1 𝑁 , so π‘Ÿ π‘š / 𝑠 = 𝑛 / 𝑑 for some 𝑑 ∈ 𝑆 and 𝑛 ∈ 𝑁 . There exists 𝑑 β€² ∈ 𝑆 such that 𝑑 β€² 𝑑 π‘Ÿ π‘š = 𝑑 β€² 𝑠 𝑛 ∈ 𝑁 . If 𝑑 β€² 𝑑 π‘Ÿ π‘š = 0 , then 𝑑 𝑑 β€² ∈ ( 0 ∢ π‘Ÿ π‘š ) ∩ 𝑆 βŠ† 𝑃 ∩ 𝑆 = βˆ… , a contradiction. So, 0 β‰  𝑑 𝑑 β€² π‘Ÿ π‘š ∈ 𝑁 and √ 𝑑 𝑑 β€² βˆ‰ ( 𝑁 ∢ 𝑀 ) , then π‘Ÿ π‘š ∈ 𝑁 . Thus, ( 𝑁 𝑃 ∢ 𝑅 𝑃 𝑀 𝑃 ) βŠ† ( 𝑁 ∢ 𝑅 𝑀 ) 𝑃 . Clearly, ( 𝑁 ∢ 𝑅 𝑀 ) 𝑃 βŠ† ( 𝑁 𝑃 ∢ 𝑅 𝑃 𝑀 𝑃 ) , so the proof is complete.

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

Proof. Let 𝐾 be a weakly primary submodule of 𝑀 𝑃 . So, 𝐾 = 𝑁 𝑃 for some submodule 𝑁 of 𝑀 . We show that 𝑁 is weakly primary submodule of 𝑀 . Let 0 β‰  π‘Ÿ π‘š ∈ 𝑁 , so 0 / 1 β‰  π‘Ÿ π‘š / 1 ∈ 𝑁 𝑃 (if π‘Ÿ π‘š / 1 = 0 / 1 , then 𝑠 π‘Ÿ π‘š = 0 for some 𝑠 ∈ 𝑆 , 𝑠 ∈ ( 0 ∢ π‘Ÿ π‘š ) ∩ 𝑆 βŠ† 𝑃 ∩ 𝑆 = βˆ… , a contradiction). Hence, ( π‘Ÿ / 1 ) 𝑛 ∈ ( 𝑁 𝑃 ∢ 𝑅 𝑃 𝑀 𝑃 ) βŠ† ( 𝑁 ∢ 𝑃 𝑀 ) 𝑃 for some positive integer 𝑛 by Lemma 2.7 or π‘š / 1 ∈ 𝑁 𝑃 , since 𝑁 𝑃 is weakly primary. Thus, π‘Ÿ 𝑛 ∈ ( 𝑁 ∢ 𝑀 ) for some positive integer 𝑛 or π‘š ∈ 𝑁 , as required. Let 𝑁 be a weakly primary submodule of 𝑀 . Then, by Proposition 2.6, 𝑁 𝑃 is weakly primary submodule of 𝑀 𝑃 .

3. Unions of Weakly Primary Submodules

Definition 3.1. Let 𝑀 be a module over a commutative ring 𝑅 and 𝑁 a submodule of 𝑀 ; 𝑁 is called a 𝑒 𝑀 𝑝 π‘Ÿ -submodule of 𝑀 , provided that 𝑁 contained in a finite weakly primary submodules of 𝑀 must be contained in one of those weakly primary submodules. 𝑀 is called a 𝑒 𝑀 𝑝 π‘Ÿ -module if every submodule of 𝑀 is a 𝑒 𝑀 𝑝 π‘Ÿ -submodule.

Clearly, every 𝑒 -module is 𝑒 𝑀 𝑝 π‘Ÿ -module, and every 𝑒 𝑀 𝑝 π‘Ÿ -module is 𝑒 𝑝 π‘Ÿ -module and also, every 𝑒 𝑝 π‘Ÿ -module is 𝑒 π‘š -module.

Theorem 3.2. Let 𝑀 be a module over a commutative ring 𝑅 and 𝐾 a submodule of 𝑀 . Then,(i)if 𝑁 is a 𝑒 𝑀 𝑝 π‘Ÿ -submodule of 𝑀 and 𝐾 is a weakly primary submodule of 𝑀 such that 𝐾 βŠ† 𝑁 , then 𝑁 / 𝐾 is a 𝑒 𝑀 𝑝 π‘Ÿ -submodule of 𝑅 -module 𝑀 / 𝐾 ,(ii)if 𝑁 / 𝐾 is a 𝑒 𝑀 𝑝 π‘Ÿ -submodule of 𝑀 / 𝐾 , then 𝑁 is a 𝑒 𝑀 𝑝 π‘Ÿ -submodule of 𝑀 .

Proof. (i) Let 𝑁 / 𝐾 βŠ† 𝐿 1 βˆͺ 𝐿 2 βˆͺ β‹― βˆͺ 𝐿 𝑛 , where 𝐿 𝑖 ’s are weakly primary submodules of 𝑀 / 𝐾 . Then, by Lemma 2.3, there exists weakly primary submodules 𝑁 𝑖 of 𝑀 such that 𝐿 𝑖 = 𝑁 𝑖 / 𝐾 for 𝑖 = 1 , 2 , … , 𝑛 . So, 𝑁 / 𝐾 βŠ† 𝑁 1 / 𝐾 βˆͺ β‹― βˆͺ 𝑁 𝑛 / 𝐾 = 𝑁 1 βˆͺ β‹― βˆͺ 𝑁 𝑛 / 𝐾 , so 𝑁 βŠ† 𝑁 1 βˆͺ 𝑁 2 βˆͺ β‹― βˆͺ 𝑁 𝑛 . Hence, 𝑁 βŠ† 𝑁 π‘˜ for some π‘˜ , since 𝑁 is 𝑒 𝑀 𝑝 π‘Ÿ -submodule. Thus 𝑁 / 𝐾 βŠ† 𝑁 π‘˜ / 𝐾 , for some π‘˜ , as needed.
(ii) Let 𝑁 βŠ† 𝑁 1 βˆͺ 𝑁 2 βˆͺ β‹― βˆͺ 𝑁 𝑛 , where 𝑁 𝑖 ’s are weakly primary submodule of 𝑀 . So, 𝑁 / 𝐾 βŠ† 𝑁 1 βˆͺ β‹― βˆͺ 𝑁 𝑛 / 𝐾 = 𝑁 1 / 𝐾 βˆͺ β‹― βˆͺ 𝑁 𝑛 / 𝐾 . Therefore, 𝑁 𝑖 / 𝐾 ’s are weakly primary submodules of 𝑀 / 𝐾 , so N / 𝐾 βŠ† 𝑁 π‘˜ / 𝐾 for some π‘˜ . Hence, 𝑁 βŠ† 𝑁 π‘˜ for some π‘˜ , as required.

Theorem 3.3. Let 𝑀 be an 𝑅 -module, 𝑁 a weakly primary submodule of 𝑀 , and 𝑀 a 𝑒 𝑀 𝑝 π‘Ÿ -module. Then, 𝑀 / 𝑁 is a 𝑒 𝑀 𝑝 π‘Ÿ -module.

Proof. By Theorem 3.2.

Theorem 3.4. Let 𝑀 be a module over quasi local ring ( 𝑅 , 𝑃 ) . Then, 𝑀 is a 𝑒 𝑀 𝑝 π‘Ÿ -module if and only if 𝑅 𝑃 module 𝑀 𝑃 is a 𝑒 𝑀 𝑝 π‘Ÿ -module.

Proof. Let 𝑀 be a 𝑒 𝑀 𝑝 π‘Ÿ -module, and let 𝐾 be a submodule of 𝑅 𝑃 -module 𝑀 𝑃 such that 𝐾 βŠ† 𝐾 1 βˆͺ β‹― βˆͺ 𝐾 𝑛 , where 𝐾 𝑖 ’s are weakly primary submodules of 𝑀 𝑃 . So, 𝐾 = 𝑁 𝑃 for some submodule 𝑁 of 𝑀 and 𝐾 𝑖 = ( 𝑁 𝑖 ) 𝑃 for some weakly primary submodules 𝑁 i of 𝑀 by Theorem 2.8. Hence, 𝑁 𝑃 βŠ† ( 𝑁 1 ) 𝑃 βˆͺ β‹― βˆͺ ( 𝑁 𝑛 ) 𝑃 , so 𝑁 𝑃 βŠ† ( 𝑁 1 βˆͺ β‹― βˆͺ 𝑁 𝑛 ) 𝑃 , thus 𝑁 βŠ† 𝑁 1 βˆͺ β‹― βˆͺ 𝑁 𝑛 by [1, Theorem  2.8]. Therefore, by hypothesis 𝑁 βŠ† 𝑁 π‘˜ for some 1 ≀ π‘˜ ≀ 𝑛 . So 𝑁 𝑃 βŠ† ( 𝑁 π‘˜ ) 𝑃 for some 1 ≀ π‘˜ ≀ 𝑛 as needed.
Conversely, let 𝑅 𝑃 -module 𝑀 𝑃 is a 𝑒 𝑀 𝑝 π‘Ÿ -module, and let 𝑁 be a submodule of 𝑀 such that 𝑁 βŠ† 𝑁 1 βˆͺ β‹― βˆͺ 𝑁 𝑛 , where 𝑁 𝑖 ’s are weakly primary submodules of 𝑀 . So, 𝑁 𝑃 βŠ† ( 𝑁 1 βˆͺ β‹― βˆͺ 𝑁 𝑛 ) 𝑃 = ( 𝑁 1 ) 𝑃 βˆͺ β‹― βˆͺ ( 𝑁 𝑛 ) 𝑃 . Thus, 𝑁 𝑃 βŠ† ( 𝑁 π‘˜ ) 𝑃 for some 1 ≀ π‘˜ ≀ 𝑛 by hypothesis. Then, 𝑁 βŠ† 𝑁 π‘˜ for some 1 ≀ π‘˜ ≀ 𝑛 . So 𝑀 is a 𝑒 𝑀 𝑝 π‘Ÿ -module.

Theorem 3.5. Let 𝑀 be a finitely generated 𝑅 -module. Then, 𝑀 is a 𝑒 π‘š -module if and only if every submodule 𝑁 in 𝑀 such that ⋃ 𝑁 βŠ† 𝑛 𝑖 = 1 𝑃 𝑖 , where 𝑃 𝑖 ’s are weakly primary submodules implies that 𝑁 + 𝑃 𝑖 β‰  𝑀 for some 𝑖 .

Proof. Let 𝑀 be a finitely generated 𝑒 π‘š -module. Suppose that 𝑁 be a submodule of 𝑀 such that ⋃ 𝑁 βŠ† 𝑛 𝑖 = 1 𝑃 𝑖 , where 𝑃 𝑖 ’s are weakly primary submodules of 𝑀 . For each 𝑃 𝑖 , 𝑀 𝑖 is a maximal submodule containing 𝑃 𝑖 . Then, ⋃ 𝑁 βŠ† 𝑛 𝑖 = 1 𝑀 𝑖 , and so, 𝑁 βŠ† 𝑀 𝑖 for some 𝑖 by hypothesis. Since 𝑃 𝑖 βŠ† 𝑀 𝑖 , we have 𝑁 + 𝑃 𝑖 βŠ† 𝑀 𝑖 β‰  𝑀 .
Conversely, let 𝑁 be a submodule of 𝑀 such that ⋃ 𝑁 βŠ† 𝑛 𝑖 = 1 𝑀 𝑖 , where 𝑀 𝑖 ’s are maximal submodules of 𝑀 . Since every maximal submodule is weakly primary submodule, then 𝑁 + 𝑀 𝑖 β‰  𝑀 for some 𝑖 by hypothesis. Therefore, since 𝑀 𝑖 βŠ† 𝑁 + 𝑀 𝑖 / β«… 𝑀 , then 𝑁 + 𝑀 𝑖 = 𝑀 𝑖 , so 𝑁 βŠ† 𝑀 𝑖 for some 𝑖 . The proof is complete.

Definition 3.6. By a chain of weakly primary submodules of an 𝑅 -module 𝑀 , we mean a finite strictly increasing sequence 𝑃 1 βŠ† β‹― βŠ† 𝑃 𝑛 ; the weakly primary dimension of this chain is 𝑛 . We define the weakly primary dimension of 𝑀 to be the supremum of the lengths of all chains of weakly primary submodules in 𝑀 .

Theorem 3.7. Let 𝑀 be a finitely generated 𝑅 -module with weakly primary dimension 1. Then, 𝑀 is a 𝑒 𝑀 𝑝 π‘Ÿ -module if and only if 𝑀 is a 𝑒 π‘š -module.

Proof. Let 𝑀 be a 𝑒 𝑀 𝑝 π‘Ÿ -module. Since every 𝑒 𝑀 𝑝 π‘Ÿ -module is a 𝑒 π‘š -module, so 𝑀 is a 𝑒 π‘š -module.
Conversely, let 𝑀 be a 𝑒 π‘š -module. Let 𝑁 be a nonzero submodule of 𝑀 such that 𝑁 βŠ† 𝑃 1 ⋃ 𝑃 2 ⋃ β‹― ⋃ 𝑃 𝑛 , where 𝑃 𝑖 ’s are weakly primary submodules of 𝑀 . We may assume that 𝑃 𝑖 β‰  0 for all 𝑖 ∈ { 1 , 2 , … , 𝑛 } . By Theorem 3.5, 𝑁 + 𝑃 𝑖 β‰  𝑀 for some 𝑖 . There exists a maximal submodule 𝑃 𝑖 of 𝑀 such that 𝑁 + 𝑃 𝑖 βŠ† 𝑀 𝑖 . Since 0 is a weakly primary submodule of 𝑀 , so we have 0 / β«… 𝑁 𝑖 βŠ† 𝑀 𝑖 . Hence, since weakly primary dimension of 𝑀 is 1; 𝑃 𝑖 = 𝑀 𝑖 . Consequently, 𝑁 βŠ† 𝑃 𝑖 for some 𝑖 , as needed.

Theorem 3.8. Let 𝑀 be an 𝑅 -module with 𝑇 ( 𝑀 ) = 0 . Then, 𝑀 is a 𝑒 𝑝 π‘Ÿ -module if and only if 𝑀 is a 𝑒 𝑀 𝑝 π‘Ÿ -module.

Proof. Let 𝑀 be a 𝑒 𝑀 𝑝 π‘Ÿ -module. Then, 𝑀 is a 𝑒 𝑝 π‘Ÿ -module since every primary submodule is weakly primary. Let 𝑀 be a 𝑒 𝑝 -module and 𝑁 a submodule of 𝑀 such that 𝑁 βŠ† 𝑃 1 ⋃ 𝑃 2 ⋃ β‹― ⋃ 𝑃 𝑛 where 𝑃 𝑖 ’s are weakly primary submodules of 𝑀 . By Proposition 2.1, 𝑃 𝑖 ’s are primary submodule, so 𝑁 βŠ† 𝑃 𝑖 for some 𝑖 , as needed.

References

  1. F. Farzalipour and P. Ghiasvand, β€œQuasi multiplication modules,” Thai Journal of Mathematics, vol. 2, pp. 361–366, 2009.