On modules satisfying the descending chain condition on r-submodules

Anebri A., Mahdou N., Tekir Ü.

Communications In Algebra, vol.49, pp.1-9, 2021 (SCI-Expanded)

  • Publication Type: Article / Article
  • Volume: 49
  • Publication Date: 2021
  • Doi Number: 10.1080/00927872.2021.1958828
  • Journal Name: Communications In Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.1-9
  • Marmara University Affiliated: Yes

Abstract

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, we introduce the concept of r-Artinian modules which is a new generalization of Artinian modules. An R-module M is called an r-Artinian module if M satisfies the descending chain condition on r-submodules. Also, we call the ring R to be an r-Artinian ring if R is an r-Artinian R-module. We prove that an R-module M is an r-Artinian module if and only if its total quotient module is an Artinian module. In particular, we observe that r-Artinian modules generalize S-Artinian modules, for some particular multiplicatively closed subsets S of R. Also, we extend many properties of Artinian modules to r-Artinian modules. Finally, we use the idealization construction to give non-trivial examples of r-Artinian rings that are not Artinian.