  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 $D,D'$ are Dedekind
domains and there is an order isomorphism between $\mathcal{P}(D)$ and
$\mathcal{P}(D')$, then the rank of the class groups of $D$ and $D'$ is the
same.

Commutative Algebra