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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 939687, 5 pages
doi:10.5402/2011/939687
Research Article

Notes on the Union of Weakly Primary Submodules

Peyman Ghiasvand and Farkhonde Farzalipour

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. Rozikov

Copyright © 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 ≠ 𝑟 𝑚 ∈ 𝑁 for 𝑟 ∈ 𝑅 and 𝑚 ∈ 𝑀 , which implies that either 𝑚 ∈ 𝑁 or 𝑟 𝑛 𝑀 ⊆ 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.