Radicals of principal ideals and the class group of a Dedekind domain

Abstract

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D^{\prime }$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and $\mathcal{P}(D^{\prime })$, then the rank of the class groups of $D$ and $D^{\prime }$ is the same.