Fuchs' problem for indecomposable abelian groups

TL;DR

This paper classifies indecomposable abelian groups that can be realized as the group of units of a commutative ring across all characteristics, addressing a long-standing open problem posed by Fuchs.

Contribution

It provides a complete classification of indecomposable abelian groups that are realizable as unit groups of rings, advancing understanding in algebraic structures.

Findings

Identifies which indecomposable abelian groups are realizable as unit groups.

Classifies these groups according to the characteristic of the ring.

Resolves a 50-year-old open question posed by Fuchs.

Abstract

More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. We provide an answer to this question in the case of indecomposable abelian groups by classifying the indecomposable abelian groups that are realizable as the group of units of a ring of any given characteristic.