On Cohen's theorem for Artinian modules

TL;DR

This paper extends Cohen's theorem to finitely embedded modules over rings, characterizing Artinian modules via prime ideals and submodule conditions.

Contribution

It proves a new necessary and sufficient condition for finitely embedded modules to be Artinian, generalizing Cohen's theorem.

Findings

Characterization of Artinian modules via prime ideals

Existence of specific submodules with finitely embedded quotients

Extension of Cohen's theorem to finitely embedded 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 $M[\mathfrak{p}]\subseteq N^\mathfrak{p}\subseteq (0:_M\mathfrak{p})$.