Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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COMMUTATIVE RINGS AND MODULES THAT ARE r-NOETHERIAN

  • Anebri, Adam (Laboratory of Modelling and Mathematical Structures Department of Mathematics Faculty of Science and Technology of Fez University S.M. Ben Abdellah Fez) ;
  • Mahdou, Najib (Laboratory of Modelling and Mathematical Structures Department of Mathematics Faculty of Science and Technology of Fez University S.M. Ben Abdellah Fez) ;
  • Tekir, Unsal (Department of Mathematics Marmara University)
  • Received : 2020.10.18
  • Accepted : 2021.03.10
  • Published : 2021.09.30
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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.

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References

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