The Ordered Group of Invertible Ideals of a Prüfer Domain of Finite Character

Abstract

Abstract Let D be a Prüfer domain, and denote by ± bℑ(D) the multiplicative group of all invertible fractional ideals of D, ordered by A ≤ B if and only if A ⊇ B. Denote by G i the value group of the valuation associated with the valuation ring D M i , where {M i } i∈I is the collection of all maximal ideals of D. In this note we prove that the natural map from ± bℑ(D) into ± b∏ i∈I G i is an isomorphism onto the cardinal sum ± b∐ i∈I G i if and only if D is h-local. As a corollary, the group of divisibility of an h-local Bézout domain is isomorphic to ± b∐ i∈I G i , the notation being as above.