Of Integral Domains

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 [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]. We also characterize GCD domains with Boolean t-class semigroup, recovering thus recent results on Bezout domains [32].