Clifford Regular Domains

Abstract

Ž .The class semigroup of a commutative integral domain R is the semigroup S S R of the isomorphism classes of the nonzero ideals of R with operation induced by Ž . multiplication. A domain R is said to be Clifford regular if S S R is a Clifford Ž . semigroup, i.e. S S R is the disjoint union of the subgroups associated to the idempotent elements. In this paper we characterize the noetherian and the integrally closed Clifford regular domains and find some properties of an arbitrary Clifford regular domain. ᮊ

Bases of Pre-Riesz Groups and Conrad’s F-Condition

“…Recently, it has been recalled in [27] to answer a question posed by Bazzoni in [4,5]. Next, the case of Inv * (D), with * of finite character, being a Riesz group satisfying Conrad's F-condition has been studied by Dumitrescu and Zafrullah [9] for the so-called t-Schreier domains.…”

Finite character, local stability property and local invertibility property

“…Zanardo and Zannier proved that all orders in quadratic fields are Clifford regular domains [20] while Bazzoni and Salce showed that all valuation domains are Clifford regular [5]. The study of Clifford regular domains was then carried on by S. Bazzoni [1,2,3,4].…”

“…Since [3,Theorem 4.5]. To this end, she established an interesting relation between Clifford regularity and the local invertibility property: a domain has the local invertibility property (LIP) if each ideal I that is locally invertible (i.e., IR M is invertible, for each M ∈ Max(R)) is indeed invertible.…”

“…To this end, she established an interesting relation between Clifford regularity and the local invertibility property: a domain has the local invertibility property (LIP) if each ideal I that is locally invertible (i.e., IR M is invertible, for each M ∈ Max(R)) is indeed invertible. Bazzoni proved that a Clifford regular domain has the local invertibility property [3, Lemma 4.2] and conjectured that a Prüfer domain with the local invertibility property be of finite character [3,Question 6.2]. This conjecture was then proved in [11] and the local invertibility property and other related properties were later investigated by several authors [12,18,6,7].…”

Artin’s Equivalence Generalized

“…stable domains (cf. Bazzoni [2], Olberding [11], Sally-Vasconceles [13]), where the non-zero fractional ideals form a Clifford semigroup under multiplication (a Clifford semigroup is a disjoint union of groups). Artin has introduced an equivalence relation on the set of non-zero fractional ideals such that in certain integral domains the equivalence classes formed a group under multiplication.…”