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Groups in the class semigroup of a Prüfer domain of finite character. (English) Zbl 0997.13005

Summary: The class semigroup of a commutative integral domain \(R\) is the semigroup \({\mathcal S}(R)\) of the isomorphism classes of the non-zero ideals of \(R\) with operation induced by multiplication. We consider Prüfer domains of finite character, i.e. Prüfer domains in which every non-zero ideal is contained but in a finite number of maximal ideals. We proved [S. Bazzoni, J. Algebra 184, No. 2, 613-631 (1996; Zbl 0856.13014)] that, if \(R\) is such a Prüfer domain, then the semigroup \({\mathcal S}(R)\) is a Clifford semigroup, namely it is the disjoint union of the subgroups associated to each idempotent element. We gave a description of a generating set for the \(\wedge\)-semilattice of the idempotent elements of \({\mathcal S}(R)\) [in: S. Bazzoni, Abelian groups, module theory and topology, Lect. Notes Pure Appl. Math. 201, 79-89 (1998; Zbl 0928.13011)].
In the paper under review, we consider the constituent groups of the class semigroup. We prove that the groups associated to idempotent elements of \({\mathcal S}(R)\) 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.

MSC:

13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20M20 Semigroups of transformations, relations, partitions, etc.
13C20 Class groups
13G05 Integral domains
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