Najib Mahdou

Global Gorenstein dimensions

In this paper, we prove that the global Gorenstein projective dimension of a ring R is equal to the global Gorenstein injective dimension of R and that the global Gorenstein flat dimension of R is smaller than the common value of the terms of this equality.

ON COSTA'S CONJECTURE

Fn ! Fn 1 ! ! F0 ! M ! 0 of R-modules in which each Fi is finitely generated and free. In particular, 0-presented and 1-presented R-modules are finitely generated and finitely presented R-modules respectively. Let n; d 0 be integers. We say that R is a strong n-coherent ring if each n-presented R-module is ðn þ 1Þ-presented. A ring R is called an ðn; dÞ-ring if every R-module having a finite n-presentation has projective dimension at most d. An integral domain with this property will be called an ðn; dÞ-domain. For example, the ðn; oÞ-domains are the fields, the (0, 1)-domains are the Dedekind domains, and the (1, 1)-domains are the Prüfer domains [Co]. Every ðn; dÞ-ring is strong ðsupfn; dgÞ-coherent and every ðn; dÞ-domain is strong ðsupfn; d 1gÞ-coherent [Co, Theorem 2.2]. We call a commutative ring an n-Von Neumann regular ring if it is an ðn; 0Þ-ring. Thus, the 1-Von Neumann regular rings are the Von Neumann regular rings [Co, Theorem 1.3]. Later, we will give an example of a 2-Von Neumann regular ring which is not a Von Neumann regular ring.

Trivial Extensions Defined by Coherent-like Conditions

Abstract This paper investigates coherent-like conditions and related properties that a trivial extension R ≔ A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition.

On 2-von Neumann Regular Rings #

In this article, we consider 2-von Neumann regular rings, that is, rings R with the property that, if F 2 → F 1 → F 0 → E → 0 is an exact sequence of R-modules with F 0, F 1, and F 2 finitely generated free modules, then the module E is projective. For each positive integer m, as well as for m = ∞, we exhibit a class of 2-von Neumann regular rings with Krull dimension m. For this purpose, we study trivial extensions of local rings by infinite-dimensional vector spaces over their residue fields. The article includes a brief discussion of the scope and precision of our results.

First, Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.

A GENERALIZATION OF STRONGLY GORENSTEIN PROJECTIVE MODULES

This paper generalize the idea of the authors in J. Pure Appl. Algebra210 (2007) 437–445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given.

Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions.

On (Strongly) Gorenstein Von Neumann Regular Rings

In this article, we introduce and study the rings over which every module is (strongly) Gorenstein flat, which we call them (strongly) Gorenstein Von Neumann regular rings.

Gorenstein Global Dimensions and Cotorsion Dimension of Rings

In this article, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.

Gaussian Polynomials and Content Ideal in Pullbacks

This article deals mainly with rings (with zerodivisors) in which regular Gaussian polynomials have locally principal contents. Precisely, we show that if (T,M) is a local ring which is not a field, D is a subring of T/M such that qf(D) = T/M, h: T → T/M is the canonical surjection and R = h −1(D), then if T satisfies the property “every regular Gaussian polynomial has locally principal content,” then also R verifies the same property. We also show that if D is a Prüfer domain and T satisfies the property “every Gaussian polynomial has locally principal content”, then R satisfies the same property. The article includes a brief discussion of the scopes and limits of our result.