Non-Abelian Nil Groups

An abelian group G is called a nil group if the only ring R with additive group R+ = G is the zero-ring, R2 = {0}. Similarly, a non-abelian group G is defined to be a nil group if the only distributive near-ring R with R+ = G is the zero-near-ring, R2 = {0}. Several results concerning nil groups are obtained, including conditions for a finitely generated group to be nil, and a description of the nil finitely generated torsion groups. It is shown that perfect groups are nil, but non-trivial one-relator groups are not nil. For certain groups G, a complete description of the distributive near-rings R with R+ = G is given.