Groups in the class semigroup of a prüfer domain of finite character

Abstract

The class semigroup of a commutative integral domain R is the semi­group S(R) of the isomorphism classes of the nonzero ideals of R with operation induced by multiplication. We consider Prufer domains of finite character, i.e. Prüfer domains in which every nonzero ideal is contained but in a finite number of maximal ideals. In [1] it is proved that, if R is such a Prüfer domain, then the semigroup S(Ris a Clifford semigroup, namely it is the disjoint union of the subgroups associated to each idempotent element. In [2] we gave a description of a generating set for the A-semilattice of the idempotent elements of S(R). In this paper we consider the constituent groups of the class semigroup. We prove that the groups associated to idempotent elements of S(R) are extensions of class groups of overrings of (R) by means of direct products of archimedean groups of localizations of(R) at idempotent prime ideals.