%0 Journal Article
%T On a question concerning the Cohen's theorem
%J Journal of Algebra and Related Topics
%I University of Guilan
%Z 2345-3931
%A Pourmortazavi, S. S.
%A Keyvani, S.
%D 2023
%\ 06/01/2023
%V 11
%N 1
%P 49-53
%! On a question concerning the Cohen's theorem
%K Noetherian modules
%K Cohen's theorem
%K $X$-injective
%R 10.22124/jart.2022.22922.1432
%X 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.
%U https://jart.guilan.ac.ir/article_6308_acf6ba066d294be0dfbf50fc5dbe6e30.pdf