On weakly classical primary submodules

Abstract

In this paper all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a {\it classical primary submodule}, if for each $m\in M$ and elements $a,b\in R$, $abm\in N$ implies that either $am\in N$ or $b^tm\in N$ for some $t\geq1$. We introduce the notion of ``weakly classical primary submodules''. A proper submodule $N$ of $M$ is a {\it weakly classical primary submodule} if whenever $a,b\in R$ and $m\in M$ with $0\neq abm\in N$, then either $am\in N$ or $b^tm\in N$ for some $t\geq1$.