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Najib Mahdou,El Houssaine Oubouhou 대한수학회 2024 대한수학회논문집 Vol.39 No.1
Let $R$ be a commutative ring with identity. If the nilpotent radical $Nil(R)$ of $R$ is a divided prime ideal, then $R$ is called a $\phi$-ring. Let $R$ be a $\phi$-ring and $S$ be a multiplicative subset of $R$. In this paper, we introduce and study the class of nonnil-$S$-coherent rings, i.e., the rings in which all finitely generated nonnil ideals are $S$-finitely presented. Also, we define the concept of $\phi$-$S$-coherent rings. Among other results, we investigate the $S$-version of Chase's result and Chase Theorem characterization of nonnil-coherent rings. We next study the possible transfer of the nonnil-$S$-coherent ring property in the amalgamated algebra along an ideal and the trivial ring extension.
S-NOETHERIAN IN BI-AMALGAMATIONS
Kim, Hwankoo,Mahdou, Najib,Zahir, Youssef Korean Mathematical Society 2021 대한수학회보 Vol.58 No.4
This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the S-Noetherian property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. Our results allow us to construct new and original examples of rings satisfying the S-Noetherian property.
A Characterization of Nonnil-Projective Modules
김환구,Najib Mahdou,El Houssaine Oubouhou 경북대학교 자연과학대학 수학과 2024 Kyungpook mathematical journal Vol.64 No.1
Recently, Zhao, Wang, and Pu introduced and studied new concepts of nonnil-commutative diagrams and nonnil-projective modules. They proved that an R-module that is nonnil-isomorphic to a projective module is nonnil-projective, and they proposed the following problem: Is every nonnil-projective module nonnil-isomorphic to some projective module? In this paper, we delve into some new properties of nonnil-commutative diagrams and answer this problem in the affirmative.
ON WEAKLY QUASI n-ABSORBING SUBMODULES
Issoual, Mohammed,Mahdou, Najib,Moutui, Moutu Abdou Salam Korean Mathematical Society 2021 대한수학회보 Vol.58 No.6
Let R be a commutative ring with 1 ≠ 0, n be a positive integer and M be an R-module. In this paper, we introduce the concept of weakly quasi n-absorbing submodule which is a proper generalization of quasi n-absorbing submodule. We define a proper submodule N of M to be a weakly quasi n-absorbing submodule if whenever a ∈ R and x ∈ M with 0 ≠ a<sup>n</sup> x ∈ N, then a<sup>n</sup> ∈ (N :<sub>R</sub> M) or a<sup>n-1</sup> x ∈ N. We study the basic properties of this notion and establish several characterizations.
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.
The dimension of the maximal spectrum of some ring extensions
Rachida El Khalfaoui,Najib Mahdou 대한수학회 2023 대한수학회논문집 Vol.38 No.4
Let $A$ be a ring and $\J = \{\text{ideals $I$ of $A$} \,|\, J(I) = I\}$. The Krull dimension of $A$, written $\dim A$, is the sup of the lengths of chains of prime ideals of $A$; whereas the dimension of the maximal spectrum, denoted by $\dim_\J A$, is the sup of the lengths of chains of prime ideals from $\J$. Then $\dim_{\J} A\leq \dim A$. In this paper, we will study the dimension of the maximal spectrum of some constructions of rings and we will be interested in the transfer of the property $J$-Noetherian to ring extensions.
When every finitely generated regular ideal is finitely presented
Mohamed Chhiti,Salah Eddine Mahdou 대한수학회 2024 대한수학회논문집 Vol.39 No.2
In this paper, we introduce a weak version of coherent that we call regular coherent property. A ring is called regular coherent, if every finitely generated regular ideal is finitely presented. We investigate the stability of this property under localization and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions, pullbacks and amalgamated. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.
On graded $N$-irreducible ideals of commutative graded rings
Anass Assarrar,Najib Mahdou 대한수학회 2023 대한수학회논문집 Vol.38 No.4
Let $R$ be a commutative graded ring with nonzero identity and $n$ a positive integer. Our principal aim in this paper is to introduce and study the notions of graded $n$-irreducible and strongly graded $n$-irreducible ideals which are generalizations of $n$-irreducible and strongly $n$-irreducible ideals to the context of graded rings, respectively. A proper graded ideal $I$ of $R$ is called graded $n$-irreducible (respectively, strongly graded $n$-irreducible) if for each graded ideals $I_{1}, \ldots,I_{n+1}$ of $R$, $I=I_{1} \cap \cdots \cap I_{n+1}$ (respectively, $I_{1} \cap \cdots \cap I_{n+1} \subseteq I$ ) implies that there are $n$ of the $I_{i}$ 's whose intersection is $I$ (respectively, whose intersection is in $I$). In order to give a graded study to this notions, we give the graded version of several other results, some of them are well known. Finally, as a special result, we give an example of a graded $n$-irreducible ideal which is not an $n$-irreducible ideal and an example of a graded ideal which is graded $n$-irreducible, but not graded $(n-1)$-irreducible.
S-COHERENT PROPERTY IN TRIVIAL EXTENSION AND IN AMALGAMATED DUPLICATION
Mohamed Chhiti,Salah Eddine Mahdou Korean Mathematical Society 2023 대한수학회논문집 Vol.38 No.3
Bennis and El Hajoui have defined a (commutative unital) ring R to be S-coherent if each finitely generated ideal of R is a S-finitely presented R-module. Any coherent ring is an S-coherent ring. Several examples of S-coherent rings that are not coherent rings are obtained as byproducts of our study of the transfer of the S-coherent property to trivial ring extensions and amalgamated duplications.
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.