Amalgamated Algebras Issued from $$\phi $$-Chained Rings and $$\phi $$-Pseudo-Valuation Rings

Khalfi, Abdelhaq El; Kim, Hwankoo; Mahdou, Najib

doi:10.1007/s41980-020-00461-y

Bull. Iran. Math. Soc.

2020

DOI: 10.1007/s41980-020-00461-y

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Amalgamated Algebras Issued from $$\phi $$-Chained Rings and $$\phi $$-Pseudo-Valuation Rings

Abdelhaq El Khalfi

Hwankoo Kim

Najib Mahdou

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2024

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Cited by 4 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 ϕ-P-flat modules and ϕ-von Neumann regular rings

Mahdou

OUBOUHOU>

2023

J. Algebra Appl.

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Let [Formula: see text] be a commutative ring with a nonzero identity and [Formula: see text] an [Formula: see text]-module. Set [Formula: see text], if [Formula: see text]-[Formula: see text] then [Formula: see text] is called a [Formula: see text]-torsion module. An [Formula: see text]-module [Formula: see text] is said to be [Formula: see text]-flat, if [Formula: see text] is an exact [Formula: see text]-sequence, for any exact sequence of [Formula: see text]-modules [Formula: see text], where [Formula: see text] is [Formula: see text]-torsion. In this paper, we study some new properties of [Formula: see text]-flat modules. Then we introduce and study the class of [Formula: see text]-[Formula: see text]-flat modules which is a generalization of [Formula: see text]-flat modules and [Formula: see text]-flat modules. Finally, we give some new characterizations of the [Formula: see text]-von Neumann regular ring and its transfer to various contexts of constructions such as the amalgamation of rings along an ideal and trivial ring extension.

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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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Amalgamated Algebras Issued from $$\phi $$-Chained Rings and $$\phi $$-Pseudo-Valuation Rings

Cited by 4 publications

References 16 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 ϕ-P-flat modules and ϕ-von Neumann regular rings

On ϕ-(weak) global dimension

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