On nonnil-coherent modules and nonnil-Noetherian modules

Haddaoui, Younes El; Kim, Hwankoo; Mahdou, Najib

doi:10.1515/math-2022-0526

Open Mathematics

2022

DOI: 10.1515/math-2022-0526

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On nonnil-coherent modules and nonnil-Noetherian modules

Younes El Haddaoui

Hwankoo Kim

Najib Mahdou

Abstract: In this article, we introduce two new classes of modules over a ϕ \phi -ring that generalize the classes of coherent modules and Noetherian modules. We next study the possible transfer of the properties of being nonnil-Noetherian rings, … Show more

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Cited by 3 publications

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Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings

Haddaoui,

Kim,

Mahdou

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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In this paper, we introduce the concept of nonnil-FP-injective dimension for both modules and rings. We explore the characterization of strongly $$\phi $$ ϕ -rings that have a nonnil-FP-injective dimension of at most one. We demonstrate that, for a nonnil-coherent, strongly $$\phi $$ ϕ -ring R, the nonnil-FP-injective dimension of R corresponds to the supremum of the $$\phi $$ ϕ -projective dimensions of specific families of R-modules. We also define self-nonnil-injective rings as $$\phi $$ ϕ -rings that act as nonnil semi-injective modules over themselves and establish the equivalence between a strongly $$\phi $$ ϕ -ring R being $$\phi $$ ϕ -von Neumann regular and R being both nonnil-coherent and self-nonnil semi-injective. Furthermore, we extend the notion of semihereditary rings to $$\phi $$ ϕ -rings, coining the term ‘nonnil-semihereditary’ to describe rings where every finitely generated nonnil ideal is u-$$\phi $$ ϕ -projective. We provide several characterizations of nonnil-semihereditary rings through various conceptual lenses. Our study also includes an investigation of the transfer of the nonnil-semihereditary property in trivial ring extensions. Additionally, we define the nonnil-FP-projective dimension for modules and rings, showing that for any strongly $$\phi $$ ϕ -ring, a nonnil-FP-projective dimension of zero is indicative of the ring being nonnil-Noetherian. We also ascertain that, for a strongly $$\phi $$ ϕ -ring R, its nonnil-FP-projective dimension is the supremum of the NFP-projective dimensions across different families of R-modules. Lastly, we provide numerous examples to illustrate our results.

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Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings

Haddaoui,

Kim,

Mahdou

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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On Strongly Nonnil-Coherent Rings and Strongly Nonnil-Noetherian Rings

Alhazmy,

Almahdi,

El Haddaoui

et al. 2024

Bull. Iran. Math. Soc.

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On ϕ-(weak) global dimension

Haddaoui

Mahdou

2023

J. Algebra Appl.

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In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called [Formula: see text]-u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On [Formula: see text]-exact sequence and [Formula: see text]-projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of [Formula: see text]-flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On [Formula: see text]-Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on [Formula: see text]-Dedekind rings and [Formula: see text]-Prüfer rings, preprint (2022), arXiv:2103. 08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly [Formula: see text]-flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https:/[Formula: see text]/arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the [Formula: see text]-injective dimension, [Formula: see text]-projective dimension and [Formula: see text]-flat dimension for modules, and also the [Formula: see text]-(weak) global dimension of rings. Then, using these dimensions, we characterize several [Formula: see text]-rings ([Formula: see text]-Prüfer, [Formula: see text]-chained, [Formula: see text]-von Neumann, etc). Finally, we study the [Formula: see text]-(weak) global dimension of the trivial ring extensions defined by some conditions.

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On nonnil-coherent modules and nonnil-Noetherian modules

Abstract: In this article, we introduce two new classes of modules over a ϕ \phi -ring that generalize the classes of coherent modules and Noetherian modules. We next study the possible transfer of the properties of being nonnil-Noetherian rings, … Show more

Cited by 3 publications

References 17 publications

Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings

Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings

On Strongly Nonnil-Coherent Rings and Strongly Nonnil-Noetherian Rings

On ϕ-(weak) global dimension

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