Constituent groups of Clifford semigroups arising from t-closure

Abstract

The t-class semigroup of an integral domain R, denoted S t (R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal tmultiplication. We recently proved that if R is a Krull-type domain [M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968) 1-27], then S t (R) is a Clifford semigroup [S. Kabbaj, A. Mimouni, t-Class semigroups of integral domains, J. Reine Angew. Math. 612 (2007) 213-229]. In this paper, we aim to describe the idempotents of S t (R) and the structure of their associated groups. We extend and recover well-known results on class semigroups of valuation domains and Prüfer domains of finite character.

Integral domains with Boolean t-class semigroup

“…Proof. We used this fact within the proof of [31,Lemma 2.3]. We reproduce here its proof for the sake of completeness.…”

“…To prove this, we first establish the following lemma, which is a re-phrasing of Statement (3) in [31,Lemma 3.1] and where we substitute the assumption "integrally closed domain" to "valuation domain." Lemma 2.14.…”

w-Stability and Cliffordw-Regularity of Polynomial Rings

“…The first attempt to extend the notion of Clifford regularity in the setting of star operations is due to Kabbaj and Mimouni, who considered the t-operation [26,27,28,29]. Then Halter-Koch, in the language of ideal systems, introduced Clifford * -regularity for star operations of finite type [22].…”

Star Stability and Star Regularity for Mori Domains

“…The first attempt to extend the notion of Clifford regularity in the setting of star operations is due to S. Kabbaj and A. Mimouni, who studied Clifford t-regularity for PvMDs and Mori domains [20,21,22,23]. Successively F. Halter-Koch, in the language of ideal systems, introduced Clifford * -regularity for any star operation of finite type [18] and the authors of this paper continued this study in [16].…”