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http://dx.doi.org/10.4134/BKMS.b200881

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.5, 2021 , pp. 1221-1233 More about this Journal
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.
Keywords
r-Noetherian module; r-Noetherian ring; r-submodule; r-ideal; weakly Noetherian module; weakly Noetherian ring; Noetherian module; Noetherian ring; idealization;
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