On Cohen's theorem for Artinian modules

Abstract

In this paper, we prove that a finitely embedded $R$-module $M$ is Artinian if and only if for every prime ideal $\mathfrak{p}$ of $R$ with $(0{:}_{R}M)\subseteq \mathfrak{p}$, there exists a submodule $N^{\mathfrak{p}}$ of $M$ such that $M/N^{\mathfrak{p}}$ is finitely embedded and $M[\mathfrak{p}]\subseteq N^{\mathfrak{p}}\subseteq (0{:}_M\mathfrak{p})$.