Salah-Eddine Kabbaj

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.

t-class semigroups of integral domains

Abstract The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prüfer v-multiplication domain (PVMD) is of Krull type (in the sense of Griffin [M. Griffin, Rings of Krull type, J. reine angew. Math. 229 (1968), 1–27]) if and only if its t-class semigroup is Clifford. This extends Bazzoni and Salces results on valuation domains [S. Bazzoni and L. Salce, Groups in the class semigroups of valuation domains, Israel J. Math. 95 (1996), 135–155] and Prüfer domains [S. Bazzoni, Class semigroup of Prüfer domains, J. Algebra 184 (1996), 613–631], [S. Bazzoni, Idempotents of the class semigroup of a Pruüfer domain of finite character, Lect. Notes. Pure Appl. Math., Dekker, 201 (1998), 79–89], [S. Bazzoni, Groups in the class semigroup of a Prüfer domain of finite character, Comm. Algebra 28 (11) (2000), 5157–5167], [S. Bazzoni, Clifford regular domains, J. Algebra 238 (2001), 703–722].

Tensor products of Cohen–Macaulay rings: solution to a problem of Grothendieck

Abstract In this paper we solve a problem, originally raised by Grothendieck, on the transfer of Cohen–Macaulayness to tensor products of algebras over a field k . 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.

Class semigroups of integral domains

This paper seeks ring-theoretic 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 semigroup; in this case, S(R) identifies with the Boolean semigroup formed of all fractional overrings of R. In Section 4, we investigate Noetherian-like settings where the Clifford and Boolean properties of S(R) coincide with (Lipman and Sally–Vasconcelos) stability conditions; a main feature is that the Clifford property forces t-locally Noetherian domains to be one-dimensional Noetherian domains. Section 5 studies the transfer of the Clifford and Boolean properties to various pullback constructions. Our results lead to new families of integral domains with Clifford or Boolean class semigroup, moving therefore beyond the contexts of integrally closed domains or Noetherian domains.

The dimension of tensor products of k-algebras arising from pullbacks

Abstract The purpose of this paper is to compute the Krull dimension of tensor products of k -algebras arising from pullbacks. We also state a formula for the valuative dimension.

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.

Commutative algebra : noetherian and non-noetherian perspectives

- Introduction.- 1. Principal-like Ideals and Related Polynomial Content Conditions (D.D. Anderson).- 2. Zero-divisor Graphs in Commutative Rings (D.F. Anderson, M. Axtell, J. Stickles).- 3. Class Semigroups and T-class Semigroups of Domains (S. Bazzoni, S. Kabbaj).- 4. Forcing Algebras, Syzygy Bundles, and Tight Closure (H. Brenner).- 5. Beyond Totally Reflexive Modules and Back: A Survey on Gorenstein Dimensions (L.W. Christensen, H.B. Foxby, H. Holm).- 6. On V-domains: A Survey (M. Fontana, M. Zafrullah).- 7. Tensor Products of Algebras Over a Field (H. Haghighi, M. Tousi, S. Yassemi).- 8. Multiplicative Ideal Theory in the Context of Commutative Monoids (F. Halter-Koch).- 9. Projectively-Full Ideals and Compositions of Consistent Systems of Rank One Discrete Valuation Rings: A Survey (W. Heinzer, J. Ratliff, D. Rush).- 10. Direct-sum Behavior of Modules Over One-dimensional Rings (R. Karr, R. Wiegand).- 11. The Defect (F.V. Kuhlmann).- 12. The Use of Ultrafilters to Study the Structure of a Commutative Ring (K.A. Loper).- 13. Integrally Closed Overrings of Two-dimensional Noetherian Domains (B. Olberding).- 14. Almost Perfect Domains and Their Modules (L. Salce).- 15. Characteristic p Methods in Characteristic Zero via Ultraproducts (H. Schoutens).- 16. Rees Valuations (I. Swanson).- 17. Weak Normality and Seminormality (M. Vitulli).

On the prime ideal structure of tensor products of algebras

Abstract 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 S-property, strong S-property, and catenarity. Its main results lead to new examples of stably strong S-rings 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 illustrate the scope and sharpness of the results.

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.

Trace Properties and Pullbacks

Abstract An integral domain is said to have the radical trace property if the trace of each noninvertible ideal is a radical ideal. The article deals with the radical trace property and related trace properties with regard to various pairs of domains with a common ideal. Necessary and sufficient conditions are determined with regard to stability of the radical trace property in a few restricted settings.