#### Vol. 314, No. 1, 2021

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

### Dario Spirito

Vol. 314 (2021), No. 1, 219–231
##### Abstract

For a Dedekind domain $D$, let $\mathsc{𝒫}\left(D\right)$ 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 $\mathsc{𝒫}\left(D\right)$ and $\mathsc{𝒫}\left({D}^{\prime }\right)$, then the rank of the class groups of $D$ and ${D}^{\prime }$ is the same.

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##### 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
##### Authors
 Dario Spirito Dipartimento di Matematica Università di Padova Padova Italy Dipartimento di Matematica e Fisica Università degli Studi “Roma Tre” Roma Italy