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S. Kabbaj

Publications and source records attributed to S. Kabbaj.

At least 19 recordsLinked to original sources
arXiv2020

Controlled $K$-Fusion Frame for Hilbert Spaces

$K$-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled $K$-fusion frames, and we develop some results on the controlled $K$-fusion frames for Hilbert spaces, which generalized some well known of controlled fusion frames case. also we discuss some characterizations of controlled Bessel $K$-fusion sequences and of controlled Bessel $K$-fusion. Further, we analyse stability conditions of controlled $K$-fusion frames under perturbation.

math.FA↗
arXiv2019

Controlled continuous g-Frames in Hilbert $C^{\ast}$-Modules

Frame Theory has a great revolution in recent years, this Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the concept of Controlled Continuous g-Frames in Hilbert $C^{\ast}$-Modules. Also we give some properties.

math.OA↗
arXiv2017

A special chain theorem for the embedding dimension

This paper establishes an analogue of the special chain theorem for the embedding dimension of polynomial rings, with direct application on the (embedding) codimension. In particular, we recover a classic result on the transfer of regularity to polynomial rings (initially proved via a combination of Serre's result on finite global dimension and Hilbert theorem on syzygies). A second application characterizes regularity in general settings of localizations of polynomial rings, including Nagata rings and Serre's conjecture rings.

math.AC↗
arXiv2017

Embedding dimension and codimension of tensor products of algebras over a field

Let k be a field. This paper investigates the embedding dimension and codimension of Noetherian local rings arising as localizations of tensor products of k-algebras. We use results and techniques from prime spectra and dimension theory to establish an analogue of the "special chain theorem" for the embedding dimension of tensor products, with effective consequence on the transfer or defect of regularity as exhibited by the (embedding) codimension.

math.AC↗
arXiv2016

Matlis' semi-regularity in trivial ring extensions issued from integral domains

This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis') semi-regular ring (also known as IF-ring) along with related concepts, such as coherence, in trivial ring extensions issued from integral domains. All along the paper, we put the new results in use to enrich the literature with new families of examples subject to semi-regularity.

math.AC↗
arXiv2016

Zaks' conjecture on rings with semi-regular proper homomorphic images

In this paper, we prove an extension of Zaks' conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result extends and recovers Levy's related result on Noetherian rings and Matlis' related result on Prufer domains. It also globalizes Couchot's related result on chained rings. New examples of rings with semi-regular proper homomorphic images stem from the main result via trivial ring extensions.

math.AC↗
arXiv2016

On t-reductions of ideals in pullbacks

This paper investigates t-reductions of ideals in pullback constructions. Section 2 examines the correlation between the notions of reduction and t-reduction in pseudo-valuation domains. Section 3 solves an open problem on whether the finite t-basic and v-basic ideal properties are distinct. We prove that these two notions coincide in any arbitrary domain. Section 4 features the main result, which establishes the transfer of the finite t-basic ideal property to pullbacks in line with Fontana-Gabelli's result on PvMDs and Gabelli-Houston's result on v-domains. This allows us to enrich the literature with new families of examples, which put the class of domains subject to the finite t-basic ideal property strictly between the two classes of v-domains and integrally closed domains.

math.AC↗
arXiv2016

Star-reductions of ideals and Prufer v-multiplication domains

Let R be a commutative ring and I an ideal of R. A sub-ideal J of I is a reduction of I if JI^n = I^n+1 for some positive integer n. The ring R has the (finite) basic ideal property if (finitely generated) ideals of R do not have proper reductions. Hays characterized (one-dimensional) Prufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays' results to Prufer v-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among star-basic properties for certain star operations.

math.AC↗
arXiv2016

t-Reductions and t-integral closure of ideals

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1} +...+ a_{n-1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of ideals and its correlation with reductions. Namely, Section 2 identifies basic properties of t-reductions of ideals and features explicit examples discriminating between the notions of reduction and t-reduction. Section 3 investigates the concept of t-integral closure of ideals, including its correlation with t-reductions. Section 4 studies the persistence and contraction of t-integral closure of ideals under ring homomorphisms. All along the paper, the main results are illustrated with original examples.

math.AC↗
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

Dimension theory of tensor products of AF-rings

AF-rings are algebras over a field k which satisfy the Altitude Formula over k. This paper surveys a few works in the literature on the Krull and valuative dimensions of tensor products of AF-rings. The first section extends Wadsworth's classical results on the Krull dimension of AF-domains to the larger class of AF-rings. It also provides formulas for computing the valuative dimension with effect on the transfer of the (locally) Jaffard property. The second section studies tensor products of AF-rings over a zero-dimensional ring. Most results on algebras over a field are extended to these general constructions. The third section establishes formulas for the Krull and valuative dimensions of tensor products of pullbacks issued from AF-domains. Throughout, examples are provided to illustrate the scope and limits of the results.

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↗
arXiv2016

Weak global dimension of Prufer-like rings

In 1969, Osofsky proved that a chained ring (i.e., local arithmetical ring) with zero divisors has infinite weak global dimension; that is, the weak global dimension of an arithmetical ring is 0, 1, or infinite. In 2007, Bazzoni and Glaz studied the homological aspects of Prufer-like rings, with a focus on Gaussian rings. They proved that Osofsky's aforementioned result is valid in the context of coherent Gaussian rings (and, more generally, in coherent Prufer rings). They closed their paper with a conjecture sustaining that ``the weak global dimension of a Gaussian ring is 0, 1, or infinite. In 2010, Bakkari-Kabbaj-Mahdou provided an example of a Gaussian ring which is neither arithmetical nor coherent and has an infinite weak global dimension. In 2011, Abuhlail-Jarrar-Kabbaj introduced and investigated the new class of fqp-rings which stands strictly between the two classes of arithmetical rings and Gaussian rings. Then, they proved the Bazzoni-Glaz conjecture for fqp-rings. This paper surveys a few recent works in the literature on the weak global dimension of Prufer-like rings making this topic accessible and appealing to a broad audience. As a prelude to this, the first section of this paper provides full details for Osofsky's proof of the existence of a module with infinite projective dimension on a chained ring. Numerous examples -arising as trivial ring extensions- are provided to illustrate the concepts and results involved in this paper.

math.AC↗
arXiv2016

Integral domains with Boolean t-Class semigroup

The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by t-multiplication. This paper investigates integral domains with Boolean t-class semigroup with an emphasis on the GCD and stability conditions. The main results establish t-analogues for well-known results on Prufer domains and Bezout domains of finite character.

math.AC↗
arXiv2014

Bi-Amalgamated algebras along ideals

Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two commutative ring homomorphisms and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \emph{bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B\times C$ given by $$A\bowtie^{f,g}(J,J'):=\big\{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big\}.$$ This paper investigates ring-theoretic properties of \emph{bi-amalgamations} and capitalizes on previous works carried on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and show how these constructions arise as pullbacks. The fourth section investigates the transfer of some basic ring theoretic properties to bi-amalgamations and the fifth section is devoted to the prime ideal structure of these constructions. All new results agree with recent studies in the literature on D'Anna-Finocchiaro-Fontana's amalgamations and duplications.

math.AC↗
arXiv2012

Regularity of tensor products of k-algebras

This paper tackles a problem on the possible transfer of regularity to tensor products of algebras over a field k. The main result establishes necessary and sufficient conditions for a Noetherian tensor product of two extension fields of k to inherit regularity in various settings of separability. Thereby, we provide some applications as well as several original examples to illustrate or delimit the scope of the established results.

math.AC↗
arXiv2010

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prufer conditions between a ring and its total ring of quotients. Section 4 examines various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prufer rings.

math.AC↗