Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2012; 49(2): 435-447

Printed March 1, 2012

https://doi.org/10.4134/JKMS.2012.49.2.435

Copyright © The Korean Mathematical Society.

Semiprime submodules of graded multiplication modules

Sang Cheol Lee and Rezvan Varmazyar

Chonbuk National University, Islamic Azad University

Abstract

Let $G$ be a group. Let $R$ be a $G$-graded commutative ring with identity and $M$ be a $G$-graded multiplication module over $R$. A proper graded submodule $Q$ of $M$ is semiprime if whenever $I^nK\subseteq Q$, where $I\subseteq h(R)$, $n$ is a positive integer, and $K\subseteq h(M)$, then $IK\subseteq Q$. We characterize semiprime submodules of $M$. For example, we show that a proper graded submodule $Q$ of $M$ is semiprime if and only if $grad(Q)\cap h(M) = Q\cap h(M)$. Furthermore if $M$ is finitely generated, then we prove that every proper graded submodule of $M$ is contained in a graded semiprime submodule of $M$. A proper graded submodule $Q$ of $M$ is said to be almost semiprime if \begin{align*} (grad (Q)\cap h(M))\backslash (grad(0_M)\cap h(M))= (Q\cap h(M))\backslash (grad(0_M)\cap Q \cap h(M)). \end{align*} Let $K,\,Q$ be graded submodules of $M$. If $K$ and $Q$ are almost semiprime in $M$ such that $Q+K\neq M$ and $Q \cap K \subseteq M_g$ for all $g \in G$, then we prove that $Q+K$ is almost semiprime in $M$.

Keywords: graded multiplication module, semiprime submodule, almost semiprime

MSC numbers: 13C13, 13A02, 16W50

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