Abdeslam Mimouni

Integral Domains in Which Each Ideal Is a W-Ideal

ABSTRACT In this paper, we investigate integral domains in which each ideal is a w-ideal (i.e. the d- and w-operations are the same), called the DW-domains. In some sense this study is similar to that one given in Houston and Zafrullah (1988) [Houston, E., Zafrullah, M. (1988). Integral domains in which each t-ideal is divisorial. Michigan Math. J. 35:291–300.] for the TV-domains. We prove that a domain R is a DW-domain if and only if each maximal ideal of R is a w-ideal, and if R is a domain such that R M is a DW-ideal for each maximal ideal M of R, then so is R, and the equivalence holds when R is v-coherent. We describe the w-operation on pull–backs in order to provide original examples. #Communicated by A. Facchini.

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].

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.

TW-domains and Strong Mori domains

Abstract In this paper we are mainly concerned with TW -domains, i.e., domains in which the w - and t -operations coincide. Precisely, we investigate possible connections with related well-known classes. We characterize the TW -property in terms of divisoriality for Mori domains and Noetherian domains. Specifically, we prove that a Mori domain R is a TW -domain if and only if R M is a divisorial domain for each t -maximal ideal M of R . It turns out that a Mori domain which is a TW -domain is a Strong Mori domain. The last section examines the transfer of the “ TW -domain” and “Strong Mori” properties to pullbacks, in order to provide some original examples.

Integral Domains Which Admit at Most Two Star Operations

In this article, we characterize domains which admit at most two star operations in the integrally closed and Noetherian cases. We also precisely count the number of star operations on an h-local Prüfer domain.

Note on Star Operations Over Polynomial Rings

This article studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a *-maximal ideal and when a *-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩ R ≠0, for a given star operation of finite character * on R[X]. We also answer negatively some questions raised by Anderson–Clarke by constructing a Prüfer domain R for which the v-operation is not stable.

On the Cardinality of Semistar Operations on Integral Domains

ABSTRACT In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the “classical” concept of star operation. In this article we establish, in finite case, relations among the number of semistar operations and the number of star operations on a domain R, the number of semistar operations on an overring T of R and the Krull dimension of R. As an application, we focus our attention to the case where R is a pullback of T obtained by the classical “D + M” constructions.

Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.

RATLIFF–RUSH CLOSURE OF IDEALS IN INTEGRAL DOMAINS

This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal

Trace Properties and Pullbacks

I