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

Dario Spirito

Vol. 314 (2021), No. 1, 219–231

DOI: 10.2140/pjm.2021.314.219

Abstract

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

Keywords

Dedekind domain, class group, radical ideals, principal ideals

Mathematical Subject Classification

Primary: 13C20, 13F05

Milestones

Received: 26 November 2020

Revised: 30 April 2021

Accepted: 16 July 2021

Published: 15 October 2021