On a question concerning the Cohen's theorem

Let R be a commutative ring with identity, and let M be an R-module.  The Cohen's theorem is the classic result that a ring is Noetherian if and only if its prime ideals are finitely generated. Parkash and Kour obtained a new version of Cohen's theorem for modules, which states that a finitely generated R-module 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 of M such that pM \subseteq N \subseteq M(p), where M(p) = \{x \in M | sx \in pM \,\,\textit{for some} \,\, s \in R \backslash p\}. In this paper, we prove this result for some classes of modules.

Keywords:

Noetherian modules , Cohen's theorem , X-injective