Fuchs' problem for dihedral groups

More than 50 years ago, László Fuchs asked which abelian groups can be the group of units of a ring. Though progress has been made, the question remains open. One could equally well pose the question for various classes of nonabelian groups. In this paper, we prove that D2,D4,D6,D8, and D12 are the only dihedral groups that appear as the group of units of a ring of positive characteristic (or, equivalently, of a finite ring), and D2 and D4k, where k is odd, are the only dihedral groups that appear as the group of units of a ring of characteristic 0.