On Dedekind domains whose class groups are direct sums of cyclic groups

TL;DR

This paper constructs Dedekind domains with class groups that are direct sums of finitely generated abelian groups, and explores their properties, including prime ideals, orders, and sets of lengths.

Contribution

It provides a method to realize Dedekind domains with prescribed class group structures and analyzes their localizations and factorization properties.

Findings

Constructed Dedekind domains with specified class groups

Established existence of prime ideals in all classes

Analyzed orders and sets of lengths in the domains

Abstract

For a given family $(G_i)_{i \in \N}$ of finitely generated abelian groups, we construct a Dedekind domain $D$ having the following properties. \begin{enumerate} \item $\Pic(D) \cong \bigoplus_{i \in \N}G_i$. \item For each $i \in \N$, there exists a submonoid $S_i \subseteq D^{\bullet}$ with $\Pic (D_{S_i}) \cong G_i$. \item Each class of $\Pic (D)$ and of all $\Pic (D_{S_i})$ contains infinitely many prime ideals. \end{enumerate} Furthermore, we study orders as well as sets of lengths in the Dedekind domain $D$ and in all its localizations $D_{S_i}$.