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Rings in which every ideal contained in the set of zero-divisors is a d-ideal
Adam Anebri,Najib Mahdou,Abdeslam Mimouni 대한수학회 2022 대한수학회논문집 Vol.37 No.1
In this paper, we introduce and study the class of rings in which every ideal consisting entirely of zero divisors is a d-ideal, considered as a generalization of strongly duo rings. Some results including the characterization of AA-rings are given in the first section. Further, we examine the stability of these rings in localization and study the possible transfer to direct product and trivial ring extension. In addition, we define the class of $d_E$-ideals which allows us to characterize von Neumann regular rings.
Adam Anebri,Najib Mahdou,Abdeslam Mimouni Korean Mathematical Society 2023 대한수학회논문집 Vol.38 No.1
In this erratum, we correct a mistake in the proof of Proposition 2.7. In fact the equivalence (3) ⇐ (4) "R is a quasi-regular ring if and only if R is a reduced ring and every principal ideal contained in Z(R) is a 0-ideal" does not hold as we only have Rx ⊆ O(S).
COMMUTATIVE RINGS AND MODULES THAT ARE r-NOETHERIAN
Anebri, Adam,Mahdou, Najib,Tekir, Unsal Korean Mathematical Society 2021 대한수학회보 Vol.58 No.5
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.