Integral Domains with Finitely Generated Groups of Divisibility

Abstract

Let $D$ be an integral domain with integral closure $\overline D$. We show that the group of divisibility $G(D)$ of $D$ is finitely generated if and only if $G(\overline D )$ is finitely generated and $\overline D /[D:\overline D ]$ is finite. We also show that $G(D)$ is finitely generated if and only if the monoid of finitely generated fractional ideals of $D$ (under multiplication) is finitely generated.