On a question concerning the Cohen's theorem

Document Type : Research Paper


1 Department of Mathematics, Guilan University, Rasht, Iran

2 Department of Mathematics, Bandar Anzali Branch, Islamic Azad University, Bandar Anzali Branch, Iran


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.