The dual notion of $r$-submodules of modules

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 an $r$-submodule if $am\in N$ with $(0:_Ma)=0$ implies that $m \in N$ for each $a\in R$ and $m\in M$. The purpose of this paper is to introduce and investigate the dual notion of $r$-submodules of $M$.