Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On Two Versions of Cohen's Theorem for Modules

  • Xiaolei Zhang (School of Mathematics and Statistics, Shandong University of Technology) ;
  • Wei Qi (School of Mathematics and Statistics, Shandong University of Technology) ;
  • Hwankoo Kim (Division of Computer Engineering, Hoseo University)
  • Received : 2021.10.16
  • Accepted : 2022.08.08
  • Published : 2023.03.31
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Abstract

Parkash and Kour obtained a new version of Cohen's theorem for Noetherian 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 N𝔭 of M such that 𝔭M ⊆ N𝔭 ⊆ M(𝔭), where M(𝔭) = {x ∈ M | sx ∈ 𝔭M for some s ∈ R \ 𝔭}. In this paper, we generalize the Parkash and Kour version of Cohen's theorem for Noetherian modules to S-Noetherian modules and w-Noetherian modules.

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Acknowledgement

The authors would like thank referees for useful comments.

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