Class Semigroups of Prüfer Domains

✍ Scribed by S Bazzoni

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 218 KB
• Volume: 184
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0279

✦ Synopsis

The class semigroup of a commutative integral domain R is the semigroup S S 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 Ž . such that the semigroup S S 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 Ž . R is invertible. We consider the ࠻ property, introduced in 1967 for Prufer domains R, stating that if ⌬ and ⌬ are two distinct sets of maximal ideals of R,

Ž . 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 C C, 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 Ž . to C C, then S S 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.