  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.
