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.
Pourmortazavi, S. S., & Keyvani, S. (2023). On a question concerning the Cohen's theorem. Journal of Algebra and Related Topics, 11(1), 49-53. doi: 10.22124/jart.2022.22922.1432
MLA
S. S. Pourmortazavi; S. Keyvani. "On a question concerning the Cohen's theorem". Journal of Algebra and Related Topics, 11, 1, 2023, 49-53. doi: 10.22124/jart.2022.22922.1432
HARVARD
Pourmortazavi, S. S., Keyvani, S. (2023). 'On a question concerning the Cohen's theorem', Journal of Algebra and Related Topics, 11(1), pp. 49-53. doi: 10.22124/jart.2022.22922.1432
VANCOUVER
Pourmortazavi, S. S., Keyvani, S. On a question concerning the Cohen's theorem. Journal of Algebra and Related Topics, 2023; 11(1): 49-53. doi: 10.22124/jart.2022.22922.1432
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