On Cohen’s theorem for modules

In this paper, we prove that if R is a commutative ring with unity and M is a finitely generated R-module, then M is Noetherian if and only if for every prime ideal P of R with $$Ann(M) \subseteq P$$ , there exists a finitely generated submodule $$N_P$$ of M such that $$PM \subseteq N_P \subseteq M(P)$$ .