Class semigroups of Prufer domains

The class semigroup of a commutative integral domain R is the semigroup L(R) of the isomorphism classes of the nonzero ideals of R with the operation induced by multiplication. The aim of this paper is to characterize the Prufer domains R such that the semigroup L(R) is a Clifford semigroup, namely a disjoint union of groups each one associated to an idempotent of the semigroup. We find a connection between this problem and the following local invertibility property: an ideal I of R is invertible if and only if every localization of I at a maximal ideal of R is invertible. We consider the (#) property, introduced in 1967 for Prufer domains R, stating that if Delta(1) and Delta(2) are two distinct sets of maximal ideals of R, then boolean AND{R(M)\M is an element of Delta(1)} not equal boolean AND{R(M)\M is an element of Delta(2)}. Let b be the class of Prufer domains satisfying the separation property (#) or with the property that each localization at a maximal ideal if finite-dimensional. We prove that, if R belongs to b, then the local invertibility property holds on R if and only if every nonzero element of R is contained only in a finite number of maximal ideals of R. Moreover if R belongs to b, then L(R) is a Clifford semigroup if and only if every nonzero element of R is contained only in a finite number of maximal ideals of R. (C) 1996 Academic Press, Inc