Every abelian group is the class group of a simple Dedekind domain

TL;DR

This paper proves that every abelian group can be realized as the class group of a noncommutative simple Dedekind domain, extending classical results to a broader noncommutative setting.

Contribution

It demonstrates that all abelian groups are class groups of noncommutative simple Dedekind domains, solving an open problem in the theory of hereditary Noetherian prime rings.

Findings

Every abelian group is realizable as a class group of a noncommutative simple Dedekind domain.

Extends classical commutative results to noncommutative algebraic structures.

Addresses an open problem posed by Levy and Robson.

Abstract

A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind domains form a second important class. We show that every abelian group is the class group of a noncommutative simple Dedekind domain. This solves an open problem stated by Levy and Robson in their recent monograph on hereditary Noetherian prime rings.