On Fuchs' Problem about the group of units of a ring

TL;DR

This paper investigates which finite groups can be realized as the group of units of a commutative ring, providing classifications for torsion-free and characteristic zero cases, and addressing related questions about group cardinalities.

Contribution

It offers a complete classification for torsion-free rings and a detailed description for characteristic zero rings, advancing understanding of Fuchs' problem.

Findings

Complete classification for torsion-free rings

Description of groups of units in characteristic zero rings

Resolution of Ditor's question on group cardinalities

Abstract

In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a previous paper \cite{DDcharp} we dealt with finite characteristic rings. In this paper we consider Fuchs' question for finite groups and we address this problem in two cases. Firstly, we study the case of torson-free rings and we obtain a complete classification of the finite groups of units which arise in this case. Secondly, we examine the case of characteristic zero rings obtaining, a pretty good description of the possible groups of units equipped with families of examples of both realizable and non-realizable groups. The main tools to deal with this general case are the Pearson and Schneider splitting of a ring \cite{PearsonSchneider70}, our previous results on finite characteristic rings \cite{DDcharp} and our classification of the groups of units of torsion-free rings. As a consequence of our results we completely answer Ditor's question \cite{ditor} on the possible cardinalities of the group of units of a ring.