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$(M)\subseteq \mathfrak{p}$,thereexistsafinitelygeneratedsubmodule$N^\mathfrak{p}$of$M$suchthat$\mathfrak{p} M\subseteq N^\mathfrak{p}\subseteq M(\mathfrak{p})$,where$M(\mathfrak{p})={x\in M\mid sx\in \mathfrak{p} M $forsome$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.