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On a question concerning the Cohen's theorem | ||
| Journal of Algebra and Related Topics | ||
| دوره 11، شماره 1، شهریور 2023، صفحه 49-53 اصل مقاله (264.16 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2022.22922.1432 | ||
| نویسندگان | ||
| S. S. Pourmortazavi1؛ S. Keyvani* 2 | ||
| 1Department of Mathematics, Guilan University, Rasht, Iran | ||
| 2Department of Mathematics, Bandar Anzali Branch, Islamic Azad University, Bandar Anzali Branch, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Noetherian modules؛ Cohen's theorem؛ $X$-injective | ||
| مراجع | ||
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