Weakly classical prime 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 classical prime submodule, if for each $m \in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$ or $bm\in N$. We introduce the concept of "weakly classical prime submodules." A proper submodule $N$ of $M$ is a weakly classical prime submodule if whenever $a,b\in R$ and $m\in M$ with $0\neq abm\in N$, then $am\in N$ or $bm\in N$.