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 tmultiplication. 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 [24]) if and only if its t-class semigroup is Clifford. This extends Bazzoni and Salce's results on valuation domains [11] and Prüfer domains [7,8,9,10].S. Kabbaj and A. Mimouni, t-Class semigroups the problem for the class of integrally closed domains by proving that "R is an integrally closed domain with Clifford class semigroup if and only if R is a Prüfer domain of finite character" [10], Theorem 4.5. It is worth recalling that, in the series of papers [39,40,41], Olberding undertook an extensive study of (Lipman and Sally-Vasconcelos) stability conditions which prepared the ground to address the correlation between stability and the theory of class semigroups.AIdeal t-multiplication converts ring notions such as PID, Dedekind, Bézout (of finite character), Prüfer (of finite character), and integrality to UFD, Krull, GCD (of finite t-character), PVMD (of finite t-character), and pseudo-integrality, respectively. Recall at this point that the PVMDs of finite t-character (i.e., each proper t-ideal is contained in only finitely many t-maximal ideals) are exactly the Krull-type rings introduced and studied by Griffin in 1967-68 [23, 24]. Also pseudo-integrality (which should be termed t-integrality) was introduced and studied in 1991 by D. F. Anderson, Houston and Zafrullah [4]. We'll provide more details about this property which turned to be crucial for our study.This paper examines ring-theoretic properties of an integral domain which reciprocally reflect in semigroup-theoretic properties of its t-class semigroup. Notions and contexts that suit best t-multiplication are studied in an attempt to parallel analogous developments and generalize well-known results on class semigroups. Recall from [10,32] that an integral domain R is Clifford regular (resp., Boole regular) if S(R) is a Clifford (resp., Boolean) semigroup. A first correlation between regularity and stability conditions can be sought through Lipman stability. Indeed, R is called an L-stable domain if n≥1 (I n : I n ) = (I : I) for every nonzero ideal I of R [1]. Lipman introduced the notion of stability in the specific setting of one-dimensional commutative semi-local Noetherian rings in order to give a characterization of Arf rings; in this context, L-stability coincides with Boole regularity [37]. By analogy, we call an integral domain R Clifford (resp., Boole) t-regular if S t (R) is a Clifford (resp., Boolean) semigroup. Clearly, a Boole t-regular domain is Clifford t-re...

Constituent groups of Clifford semigroups arising from t-closure

“…Also we shall use v 1 and t 1 to denote the v-and t-operations with respect to an overring T of R. By [26,Theorem 3.2], if R is a Krull-type domain, then S t (R) is Clifford and hence a disjoint union of subgroups G J , where J ranges over the set of idempotents of S t (R) and G J is the largest subgroup of S t (R) with unit J . Notice for convenience that in valuation and Prüfer domains the t-and trivial operations (and hence the t-class and class semigroups) coincide.…”

“…We will often appeal to some of them without explicit mention. [26,Lemma 3.3]. Now, let M be an arbitrary t-maximal ideal of R. We claim that P 2 R M = P R M .…”

“…Let F (R), Inv(R), and P (R) denote the sets of nonzero, invertible, and nonzero principal fractional ideals of R, respectively. Under this notation, the Picard group [3,4,16], class group [10,11], t-class semigroup [26], and class semigroup [9,24,25,32] of R are defined as follows: Pic(R) := Inv(R)/P (R), Cl(R) := Inv t (R)/P (R), S t (R) := F t (R)/P (R), and S(R) := F (R)/P (R). We have the settheoretic inclusions…”

Core of ideals in integral domains

“…Sally and Vasconcelos [50] used stability to settle Bass' conjecture on one-dimensional Noetherian rings with finite integral closure. Recent developments on this notion, due to Olberding [45], prepared the ground to address the correlation between stability and several concepts in multiplicative ideal theory [10,23,38]. Next, we examine the effect of (strong) stability on the core of ideals.…”

Generic formal fibers and analytically ramified stable rings

“…The main results regarding the connection between stable rings and the generic formal fiber are in Sections 5 and 6, but since our main focus is on one-dimensional stable rings, and since stable rings are also of interest in non-Noetherian commutative ring theory (for some recent examples, see [4,6,9,19,31,33,34]), we include in Sections 3 and 4 characterizations of these rings in terms of their normalization and completion.…”