# Finitely generated abelian groups of units

## TL;DR

This paper characterizes finitely generated abelian groups that can be realized as the group of units in various classes of rings, addressing Fuchs' longstanding problem with new classifications.

## Contribution

It provides a comprehensive characterization of finitely generated abelian groups as units in integral domains, torsion free, and reduced rings, extending previous partial results.

## Key findings

- Identifies which finite abelian groups occur as torsion subgroups of units.
- Determines possible ranks of unit groups in different ring classes.
- Deepens understanding of units in torsion-free rings.

## Abstract

In 1960 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 this paper we address Fuchs' question for {\it finitely generated abelian} groups and we consider the problem of characterizing those groups which arise in some fixed classes of rings $\mathcal C$, namely the integral domains, the torsion free rings and the reduced rings. To determine the realizable groups we have to establish what finite abelian groups $T$ (up to isomorphism) occur as torsion subgroup of $A^*$ when $A$ varies in $\mathcal C$, and on the other hand, we have to determine what are the possible values of the rank of $A^*$ when $(A^*)_{tors}\cong T$. Most of the paper is devoted to the study of the class of torsion-free rings, which needs a substantially deeper study.