Abstract
László 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.
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