Fuchs' problem for endomorphisms of abelian groups

TL;DR

This paper explores when the automorphisms of certain abelian groups can be realized as ring endomorphisms, focusing on specific classes like torsion-free, odd order, torsion, and finitely generated groups.

Contribution

It provides a characterization of abelian groups for which all group endomorphisms are induced by ring endomorphisms, for four key classes of groups.

Findings

Characterized when torsion-free abelian groups have endomorphism-induced ring endomorphisms

Identified conditions for groups of odd order to have this property

Analyzed torsion and finitely generated abelian groups in this context

Abstract

L\'{a}szl\'{o} Fuchs posed the following question: which abelian groups arise as the group of units in a ring? In this paper, we investigate a related question: for such realizable groups $G$, when is there a ring $R$ with unit group $G$ such that every group endomorphism of $G$ is induced by a ring endomorphism of $R$? We answer this question for four common classes of groups: torsion-free abelian groups, groups of odd order, torsion abelian groups, and finitely generated abelian groups.