### 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 _{1}∈ S

**Theorem 2.1.** Let

Conversely, suppose on the contrary that

Then by Zorn's Lemma,

We claim that

Then

So

We also claim that _{4} ∈ S

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 _{1}_{2}

Then

So

implies that

We claim that _{3}_{i}∈ N_{i}∈ M_{4}

So

Let _{j}_{5}_{j}_{i}∈ N

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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