On two versions of Cohen's theorem for modules

TL;DR

This paper extends Cohen's theorem for Noetherian modules to broader classes such as S-Noetherian and w-Noetherian modules, providing new characterizations of these modules.

Contribution

It generalizes a recent version of Cohen's theorem for Noetherian modules to S-Noetherian and w-Noetherian modules, broadening its applicability.

Findings

Generalization of Cohen's theorem to S-Noetherian modules

Extension of Cohen's theorem to w-Noetherian modules

Provides new criteria for module Noetherian properties

Abstract

Parkash and Kour obtained a new version of Cohen's theorem for Noetherian modules, which states that a finitely generated $R$-module $M$ is Noetherian if and only if for every prime ideal $\mathfrak{p}$ of $R$ with Ann$(M)\subseteq \mathfrak{p}$, there exists a finitely generated submodule $N^\mathfrak{p}$ of $M$ such that $\mathfrak{p} M\subseteq N^\mathfrak{p}\subseteq M(\mathfrak{p})$, where $M(\mathfrak{p})=\{x\in M\mid sx\in \mathfrak{p} M $ for some $s\in R \setminus \mathfrak{p} \}$. In this paper, we generalize the Parkash and Kour version of Cohen's theorem for Noetherian modules to those for $S$-Noetherian modules and $w$-Noetherian modules.