대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제58권5호
  • /
  • Pages.1221-1233
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  • 2021
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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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)
  • 투고 : 2020.10.18
  • 심사 : 2021.03.10
  • 발행 : 2021.09.30
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초록

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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참고문헌

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