On two versions of Cohen's theorem for modules

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.

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