A note on a Cohen-type 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 $(0:_RM)\subseteq \mathfrak {p}$, there exists a submodule $N^\mathfrak {p}$ of M such that $M/N^\mathfrak {p}$ is finitely embedded and $M[\mathfrak {p}]\subseteq N^\mathfrak {p}\subseteq (0:_M\mathfrak {p})$, where $M[\mathfrak {p}]=\bigcap \nolimits _{s\in R {\setminus } \mathfrak {p}}s(0:_M\mathfrak {p}).$