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

ABSTRACT

Let 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 $k[x_1,x_2,\dots ,x_n]$.

KEYWORDS:

• Bézout domain
• finite character
• group of divisibility
• Prüfer domain

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13A15
• 13F05
• 13F30
• 06F20

Acknowledgments

I would like to express the deepest gratitude to Bruce Olberding for suggesting this topic. I thank him for the helpful discussions and comments that made to achieve the aim of this article. Also, I thank referee for valuable comments and for the amount of time and effort put into reading the first draft of the paper.