On a question concerning the Cohen's theorem

Document Type : Research Paper

Authors

1 Department of Mathematics, Guilan University, Rasht, Iran

2 Department of Mathematics, Bandar Anzali Branch, Islamic Azad University, Bandar Anzali Branch, Iran

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)⊆p, there exists a finitely generated submodule N of M such that pM⊆N⊆M(p), where M(p)={x∈M|sx∈pMfor somes∈R∖p}. In this paper, we prove this result for some classes of modules. 

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References

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Journal of Algebra and Related Topics
Volume 11, Issue 1
June 2023
Pages 49-53

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