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S. Kabbaj; A. Kadri
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Let R be an integral domain and I a nonzero ideal of R. A subideal J of I is a treduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is tintegral over I if there is an equation x^{n} + a_{1}x^{n1} +...+ a_{n1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are tintegral over I is called the tintegral closure of I. This paper investigates the treductions and tintegral closure of ideals. Our objective is to...
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1602.07041
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S. Kabbaj; N. Mahdou
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This paper partly settles a conjecture of Costa on (n,d)rings, i.e., rings in which npresented 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.
Source: http://arxiv.org/abs/math/0606694v1
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K. Adarbeh; S. Kabbaj
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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') semiregular ring (also known as IFring) 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 semiregularity.
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1604.02795
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E. Houston; S. Kabbaj; A. Miomouni
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Let R be a commutative ring and I an ideal of R. A subideal 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 (onedimensional) Prufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays' results to Prufer vmultiplication domains by replacing "basic" with "wbasic," where w is...
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1602.07035
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S. Bouchiba; S. Kabbaj
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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...
Topics: Mathematics, Commutative Algebra
Source: http://arxiv.org/abs/1410.0185
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S. Kabbaj; A. Mimouni
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The tclass semigroup of an integral domain is the semigroup of fractional tideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal tmultiplication. This paper investigates ringtheoretic properties of a Noetherian domain that reflect reciprocally in the Clifford or Boolean property of its tclass semigroup.
Source: http://arxiv.org/abs/0811.4693v1
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This review paper deals with dimension theory of polynomial rings over certain families of pullbacks. While the literature is plentiful, this field is still developing and many contexts are yet to be explored. I will thus restrict the scope of the present survey, mainly, to topics I have worked on over the last decade. The set of pullback constructions studied includes D+M, D+(X_{1}, >..., X_{n})D_{S}[X_{1}, ..., X_{n}], A+XB[X], and D+I.
Source: http://arxiv.org/abs/math/0606686v1
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S. El Baghdadi; L. Izelgue; S. Kabbaj
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This paper studies the class group of graded integral domains. As an application, we state a decomposition theorem for class groups of semigroup rings. This recovers wellknown results developed for the classic contexts of polynomial rings and Krull semigroup rings. Our results are backed by original examples.
Source: http://arxiv.org/abs/math/0606693v1
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S. Kabbaj; A. Mimouni
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This paper seeks ringtheoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S(R) identifies with the Boolean semigroup formed of all fractional overrings of R. In Section 4, we...
Source: http://arxiv.org/abs/math/0606691v1
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J. Abuhlail; M. Jarrar; S. Kabbaj
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This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasiprojective. Section 2 provides some preliminaries on quasiprojective modules over commutative rings. Section 3 investigates the correlation with wellknown 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...
Source: http://arxiv.org/abs/0810.0359v2
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K. Adarbeh; S. Kabbaj
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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 Pruferlike 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...
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1601.07657
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S. Kabbaj; N. Mahdou; M. A. S. Moutui
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This paper establishes necessary and sufficient conditions for a biamalgamation 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 biamalgamations. All results are backed with new and illustrative examples arising as biamalgamations.
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1601.07653
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S. Bouchiba; S. Kabbaj
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Let k be a field. This paper investigates the embedding dimension and codimension of Noetherian local rings arising as localizations of tensor products of kalgebras. 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.
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1701.05365
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M. Chhiti; M. Jarrar; S. Kabbaj; N. Mahdou
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This paper investigates idealtheoretic 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.
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1601.07656
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S. Bouchiba; D. E. Dobbs; S. Kabbaj
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This paper is concerned with the prime spectrum of a tensor product of algebras over a field. It seeks necessary and sufficient conditions for such a tensor product to have the Sproperty, strong Sproperty, and catenarity. Its main results lead to new examples of stably strong Srings and universally catenarian rings. The work begins by investigating the minimal prime ideal structure. Throughout, several results on polynomial rings are recovered, and numerous examples are provided to...
Source: http://arxiv.org/abs/math/0606689v1
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S. Kabbaj; K. Louartit; M. Tamekkante
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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{biamalgamation} 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 ringtheoretic properties of \emph{biamalgamations} and capitalizes on...
Topics: Mathematics, Commutative Algebra
Source: http://arxiv.org/abs/1407.7074
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K. Adarbeh; S. Kabbaj
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In this paper, we prove an extension of Zaks' conjecture on integral domains with semiregular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zerodivisors). 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 semiregular proper homomorphic images stem from the main result...
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1604.03268
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S. Bouchiba; S. Kabbaj
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This paper deals with Bouvier's conjecture which sustains that finitedimensional nonNoetherian Krull domains need not be Jaffard
Source: http://arxiv.org/abs/0811.4691v1
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S. Bouchiba; F. Girolami; S. Kabbaj
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The purpose of this paper is to compute the Krull dimension of tensor products of kalgebras arising from pullbacks. We also state a formula for the valuative dimension.
Source: http://arxiv.org/abs/math/0606697v1
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S. Kabbaj; A. Mimouni
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The tclass semigroup of an integral domain R, denoted S_t(R), is the semigroup of fractional tideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal tmultiplication. We recently proved that if R is a Krulltype domain, then S_t(R) is a Clifford semigroup. This paper aims to describe the idempotents of S_t(R) and the structure of their associated groups. We extend and recover wellknown results on class semigroups of valuation domains and Prufer domains...
Source: http://arxiv.org/abs/math/0611305v3
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S. Kabbaj; A. Mimouni
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The tclass semigroup of an integral domain is the semigroup of the isomorphy classes of the tideals with the operation induced by ideal tmultiplication. This paper investigates ringtheoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its tclass semigroup. Contexts (including Lipman and SallyVasconcelos stability) that suit best tmultiplication are studied in an attempt to generalize wellknown developments on class semigroups. We...
Source: http://arxiv.org/abs/math/0606636v5
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S. Bouchiba; S. Kabbaj
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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.
Source: http://arxiv.org/abs/1202.5615v2
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S. Bouchiba; S. Kabbaj
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This paper contributes to the study of the prime spectrum and dimension theory of symbolic Rees algebra over Noetherian domains. We first establish some general results on the prime ideal structure of subalgebras of affine domains, which actually arise, in the Noetherian context, as domains between a domain $A$ and $A[a^{1}]$. We then examine closely the special context of symbolic Rees algebras (which yielded the first counterexample to the ZariskiHilbert problem). One of the results states...
Source: http://arxiv.org/abs/0903.0052v1
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C. Bakkari; S. Kabbaj; N. Mahdou
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This paper deals with wellknown 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 BazzoniGlaz conjecture on the weak dimension of Gaussian rings. Moreover, trivial ring extensions...
Source: http://arxiv.org/abs/0808.0275v2
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S. Kabbaj; N. Mahdou
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This paper investigates coherentlike 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 wellknown coherencelike results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherentlike condition.
Source: http://arxiv.org/abs/math/0606696v1
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S. Bouchiba; S. Kabbaj
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In this paper we solve a problem, originally raised by Grothendieck, on the transfer of CohenMacaulayness to tensor products of algebras over a field. As a prelude to this, we investigate the grade for some specific types of ideals that play a primordial role within the ideal structure of such constructions.
Source: http://arxiv.org/abs/math/0606692v1
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AFrings 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 AFrings. The first section extends Wadsworth's classical results on the Krull dimension of AFdomains to the larger class of AFrings. 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...
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1601.07654
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S. Kabbaj; A. Mimouni
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The tclass semigroup of an integral domain is the semigroup of the isomorphy classes of the tideals with the operation induced by tmultiplication. This paper investigates integral domains with Boolean tclass semigroup with an emphasis on the GCD and stability conditions. The main results establish tanalogues for wellknown results on Prufer domains and Bezout domains of finite character.
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1601.07660
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S. Kabbaj; A. Kadri; A. Mimouni
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This paper investigates treductions of ideals in pullback constructions. Section 2 examines the correlation between the notions of reduction and treduction in pseudovaluation domains. Section 3 solves an open problem on whether the finite tbasic and vbasic 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 tbasic ideal property to pullbacks in line with...
Topics: Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1607.06705