arXiv subjects

N. Mahdou

Publications and source records attributed to N. Mahdou.

15 recordsLinked to original sources

arXiv2016

Bi-amalgamations subject to the arithmetical property

This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations.

math.AC↗

arXiv2016

Prufer conditions in an amalgamated duplication of a ring along an ideal

This paper investigates ideal-theoretic as well as homological extensions of the Prufer 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 Prufer conditions.

math.AC↗

arXiv2015

On Armendariz-like properties in Amalgamated algebras along ideals

Let $f: A\rightarrow B$ be a ring homomorphism and $J$ be an ideal of $B$. In this paper, we investigate the transfer of Armendariz-like properties to the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by $A\bowtie^fJ)$ introduced and studied by D'Anna, Finocchiaro and Fontana in 2009. Our aim is to provide necessary and sufficient conditions for $A\bowtie^fJ,$ to be an Armendariz ring, nil-Armendariz ring and weak Armendariz ring.

math.AC↗

arXiv2009

On G-$(n,d)$-rings

The main aim of this paper is to investigate new class of rings called, for positive integers $n$ and $d$, $G-(n,d)-$rings, over which every $n$-presented module has a Gorenstein projective dimension at most $d$. Hence we characterize $n$-coherent $G-(n,0)-$rings. We conclude by various examples of $G-(n,d)-$rings.

math.AC↗

arXiv2009

Well-centered Overrings of a Commutative Ring in Pullbacks and Trivial Extensions

Let $R$ be a commutative ring with identity and $T(R)$ its total quotient ring. We extend the notion of well-centered overring of an integral domain to an arbitrary commutative ring and we investigate the transfer of this property to different extensions of commutative rings in both integral and non-integral cases. Namely in pullbacks and trivial extensions. Our aim is to provide new classes of commutative rings satisfying this property and to shed light on some open questions raised by Heinzer and Roitman in \cite{HR}.

math.AC↗

arXiv2008

On (n,d)-perfect rings

In this paper, we introduce the notion of "(n,d)-perfect rings" which is in some way a generalization of the notion of "S-rings". After we give some basic results of this rings and we survey the relationship between "A(n) property" and "(n,d)-perfect property". Finally, we investigate the "(n,d)-perfect property" in pullback rings.

math.AC↗

arXiv2008

Trivial extensions defined by Prufer conditions

This paper deals with well-known extensions of the Prufer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zerodivisors subject to various Prufer conditions. The new examples give further evidence for the validity of Bazzoni-Glaz conjecture on the weak dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.

math.AC↗

arXiv2008

On strong n-perfect rings

In this paper we introduce the notion of "strong $n$-perfect rings" which is in some way a generalization of the notion of "$n$-perfect rings". We are mainly concerned with those class of rings in the context of pullbacks. Also we exhibit a class of $n$-perfect rings that are not strong $n$-perfect rings. Finally, we establish the transfer of this notion to the direct product notions.

math.AC↗

arXiv2008

On n-Perfect Rings and Cotorsion Dimension

A ring is called $n$-perfect ($n\geq 0$), if every flat module has projective dimension less or equal than $n$. In this paper, we show that the $n$-perfectness relate, via homological approach, some homological dimension of rings. We study $n$-perfectness in some known ring constructions. Finally, several examples of $n$-perfect rings satisfying special conditions are given.

math.AC↗

arXiv2008

Gorenstein Global Dimensions and Cotorsion Dimension of Rings

In this paper, 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.

math.AC↗

arXiv2008

Rings over which all modules are strongly Gorenstein projective

One of the main results of this paper is the characterization of the rings over which all modules are strongly Gorenstein projective. We show that these kinds of rings are very particular cases of the well-known quasi-Frobenius rings. We give examples of rings over which all modules are Gorenstein projective but not necessarily strongly Gorenstein projective.

math.AC↗

arXiv2008

Global Gorenstein dimensions of polynomial rings and of direct product of rings

In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct product of rings, which gives examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.

math.AC↗

arXiv2007

Prüfer-Like Conditions in Subring Retracts and Applications

In this paper, we consider five possible extensions of the Prüfer domain notion to the case of commutative rings with zero divisors. We investigate the transfer of these Prüfer-like properties between a commutative ring and its subring retract. Our results generate new families of examples of rings subject to a given Prüfer-like conditions.

math.AC↗

arXiv2006

Trivial Extensions of Local Rings and a Conjecture of Costa

This paper partly settles a conjecture of Costa on (n,d)-rings, i.e., rings in which n-presented modules have projective dimension at most d. For this purpose, a theorem studies the transfer of the (n,d)-property to trivial extensions of local rings by their residue fields. It concludes with a brief discussion -backed by original examples- of the scopes and limits of our results.

math.AC↗

arXiv2006

Trivial Extensions Defined by Coherent-like Conditions

This paper investigates coherent-like conditions and related properties that a trivial extension might inherit from the ground ring over some classes of modules. 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.

math.AC↗