Fuchs’ Problem for Indecomposable Abelian Groups
More than 50 years ago, László 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.
Chebolu, Sunil K. and Lockridge, Keir. "Fuchs’ Problem for Indecomposable Abelian Groups." Journal of Algebra 438 (September 2015), 325-336.