@article {
author = {Pourmortazavi, S. S. and Keyvani, S.},
title = {On a question concerning the Cohen's theorem},
journal = {Journal of Algebra and Related Topics},
volume = {11},
number = {1},
pages = {49-53},
year = {2023},
publisher = {University of Guilan},
issn = {2345-3931},
eissn = {2382-9877},
doi = {10.22124/jart.2022.22922.1432},
abstract = {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},
url = {https://jart.guilan.ac.ir/article_6308.html},
eprint = {https://jart.guilan.ac.ir/article_6308_acf6ba066d294be0dfbf50fc5dbe6e30.pdf}
}