Article
Kyungpook Mathematical Journal 2023; 63(1): 29-36
Published online March 31, 2023
Copyright © Kyungpook Mathematical Journal.
On Two Versions of Cohen's Theorem for Modules
Xiaolei Zhang and Wei Qi, Hwankoo Kim∗
School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China
e-mail : zxlrghj@163.com and qwrghj@126.com
Division of Computer Engineering, Hoseo University, Asan 31499, Republic of Korea
e-mail : hkkim@hoseo.edu
Received: October 16, 2021; Revised: July 25, 2022; Accepted: August 8, 2022
Abstract
Parkash and Kour obtained a new version of Cohen's theorem for Noetherian modules, which states that a finitely generated
Keywords: Cohen's theorem,
1. Introduction
Throughout this article, all rings are commutative rings with identity and all modules are unitary. Let
Theorem. ([11, Theorem 2.1.]) Let
In the past few decades, some generalizations of Noetherian rings or Noetherian modules have been extensively studied, especially via some multiplicative subsets
2. Cohen's Theorem for S -Noetherian Modules
Let
Theorem 2.1. Let
Conversely, suppose on the contrary that
Then by Zorn's Lemma,
We claim that
Then
So
We also claim that
Thus
Let
Since
Taking
Corollary 2.2. ([11, Theorem 2.1]) Let
3. Cohen's Theorem for w -Noetherian Modules
We recall some basic knowledge on the
An
A
An
Lemma 3.1. Let
Let
Theorem 3.2. Let
Conversely, suppose on the contrary that
Since
We claim that
Then
So
implies that
We claim that
So
Let
Since
Taking
Corollary 3.3. ([15, Theorem 4.7(1)]) Let
Acknowledgements.
The authors would like thank referees for useful comments.
Footnote
The first author was supported by the National Natural Science Foundation of China (No. 12061001). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2021R1I1A3047469).
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