Some notes on nil-semicommutative rings

A ring R is defined to be nil-semicommutative if ab N(R) implies arb N(R) for a, b, r R, where N(R) stands for the set of nilpotents of R. Nil-semicommutative rings are generalization of NI rings. It is proved that (1) R is strongly regular if and only if R is von Neumann regular and nil-semicommutative; (2) Exchange nil-semicommutative rings are clean and have stable range 1; (3) If R is a nil-semicommutative right MC2 ring whose simple singular right modules are Y J-injective, then R is a reduced weakly regular ring; (4) Let R be a nil-semicommutative -regular ring. Then R is an (S, 2)-ring if and only if Z/2Z is not a homomorphic image of R. ? TBITAK.