IJPAM: Volume 31, No. 2 (2006)

ON THE PRIMARY AVOIDANCE THEOREM FOR
MODULES OVER COMMUTATIVE RINGS

Shahabaddin Ebrahimi Atani$^1$, Ünsal Tekir$^2$
$^1$Department of Mathematics
University of Guilan
P.O. Box 1914, Rashht, IRAN
e-mail: ebrahimi@guilan.ac.ir
$^2$Department of Mathematics
Marmara University
Göztepe, Ziverbey, Istanbul, 34722, TURKEY
e-mail: utekir@marmara.edu.tr


Abstract.Let $R$ be a commutative ring with identity and $M$ an $R$-module. In this paper we prove the following theorem: Let $M$ be an $R$-module $% N_{1},..., N_{n}$ be submodules of $M, $ and $N$ is a submodule of $M$ such that $N\subseteq N_{1}\bigcup N_{2}\bigcup...\bigcup N_{n}.$ Assume at most two of the $N_{k}$'$s$ are not primary submodule and $\sqrt{\left( N_{j}:M\right) }\nsubseteqq \sqrt{\left( N_{k}:M\right) }$ for every $j\neq k.$ Then $N\subseteq N_{k}$ for some $k.$

Received: July 15, 2006

AMS Subject Classification: 13C99

Key Words and Phrases: primary submodules, primary avoidance theorem

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 31
Issue: 2