A Cohen-type theorem for w-Noetherian modules

Abstract

An expanded distinguished prime w-submodule of a w-module M is defined by $M_w^e(P)=\{x\in M|sx\in {(PM)}_w\text{for}\mathrm{}\text{some}s\in R\setminus P\}$ where P is a prime w-ideal of R. Using this, we prove a Cohen-type theorem for w-Noetherian modules: A w-finite type R-module M is a w-Noetherian module if and only if for every prime w-ideal P of R with $\mathit{Ann}(M)\subseteq P,$ there exists a w-finite type w-submodule N of M such that ${(PM)}_w\subseteq N\subseteq M_w^e(P).$ As byproducts, among others, we get several conditions such that PM is a w-module where P is a prime w-ideal of R and R-module M is a w-module.

Keywords:

• Cohen’s theorem
• distinguished prime w-submodules
• w-Noetherian modules
• prime w-ideals
• prime w-submodules
• w-finite type

2020 Mathematics Subject Classification:

• 13A15
• 13E99
• 13F99

Acknowledgements

The authors would like to thank the referee for valuable suggestions and corrections, which have improved this article. Also, the authors would like to thank Professor Hwankoo Kim, Shiqi Xing, Lei Qiao for their help in improving this article.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant No.11671283).