TY - JOUR
ID - 6308
TI - On a question concerning the Cohen's theorem
JO - Journal of Algebra and Related Topics
JA - JART
LA - en
SN - 2345-3931
AU - Pourmortazavi, S. S.
AU - Keyvani, S.
AD - Department of Mathematics, Guilan University, Rasht, Iran
AD - Department of Mathematics, Bandar Anzali Branch, Islamic Azad University, Bandar Anzali Branch, Iran
Y1 - 2023
PY - 2023
VL - 11
IS - 1
SP - 49
EP - 53
KW - Noetherian modules
KW - Cohen's theorem
KW - $X$-injective
DO - 10.22124/jart.2022.22922.1432
N2 - 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.
UR - https://jart.guilan.ac.ir/article_6308.html
L1 - https://jart.guilan.ac.ir/article_6308_acf6ba066d294be0dfbf50fc5dbe6e30.pdf
ER -