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

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.

Keywords:

• h-local
• Prüfer domain
• Strong approximation theorem
• Valuation group

2000 Mathematics Subject Classification:

• 13F05
• 13F30
• 13G05
• 06F20

ACKNOWLEDGMENT

We would like to thank the referee for several very useful suggestions, especially those which helped to sharpen the results in Propositions 3 and 4.

Notes

Communicated by K. Rangaswamy.