COMMUTATIVE RINGS AND MODULES THAT ARE r-NOETHERIAN

Anebri A., Mahdou N., TEKİR Ü.

BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY, vol.58, no.5, pp.1221-1233, 2021 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 58 Issue: 5
  • Publication Date: 2021
  • Doi Number: 10.4134/bkms.b200881
  • Journal Name: BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.1221-1233
  • Keywords: r-Noetherian module, r-Noetherian ring, r-submodule, r-ideal, weakly Noetherian module, weakly Noetherian ring, Noetherian module, Noetherian ring, idealization, SUBMODULES
  • Marmara University Affiliated: Yes

Abstract

In this paper, we introduce and investigate a new class of modules that is closely related to the class of Noetherian modules. Let R be a commutative ring and M be an R-module. We say that M is an r-Noetherian module if every r-submodule of M is finitely generated. Also, we call the ring R to be an r-Noetherian ring if R is an r-Noetherian R-module, or equivalently, every r-ideal of R is finitely generated. We show that many properties of Noetherian modules are also true for r-Noetherian modules. Moreover, we extend the concept of weakly Noetherian rings to the category of modules and we characterize Noetherian modules in terms of r-Noetherian and weakly Noetherian modules. Finally, we use the idealization construction to give non-trivial examples of r-Noetherian rings that are not Noetherian.