Abstract
An expanded distinguished prime w-submodule of a w-module M is defined by 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
there exists a w-finite type w-submodule N of M such that
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.
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.