The group of divisibility of a finite character intersection of valuation rings

Abstract

ABSTRACTLet R be a Prüfer domain. The group of invertible fractional ideals ℑ(R) is an lattice-ordered group (ℓ-group) with respect to the ordering defined by A≤B if and only if B⊆A. In this work, we prove that if R has a finite character and each nonzero prime ideal of R contains a minimal nonzero prime ideal, then ℑ(R) is a cardinal sum of indecomposable semilocal ℓ-groups. We examine the ℓ-groups that can be realized as the group of invertible fractional ideals of a finite character Prüfer overring of .