Class Semigroups of Prüfer Domains Original Research Article

The class semigroup of a commutative integral domainRis the semigroupimage(R) of the isomorphism classes of the nonzero ideals ofRwith the operation induced by multiplication. The aim of this paper is to characterize the Prüfer domainsRsuch that the semigroupimage(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 idealIofRis invertible if and only if every localization ofIat a maximal ideal ofRis invertible. We consider the (#) property, introduced in 1967 for Prüfer domainsR, stating that if Δ1and Δ2are two distinct sets of maximal ideals ofR, then ∩{RMMset membership, variantΔ1}≠∩{RMMset membership, variantΔ2}. Letimagebe the class of Prüfer domains satisfying the separation property (#) or with the property that each localization at a maximal ideal if finite-dimensional. We prove that, ifRbelongs toimage, then the local invertibility property holds onRif and only if every nonzero element ofRis contained only in a finite number of maximal ideals ofR. Moreover ifRbelongs toimage, thenimage(R) is a Clifford semigroup if and only if every nonzero element ofRis contained only in a finite number of maximal ideals ofR.