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 N I 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 M C2 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 π− regular ring. Then R is an (S, 2) -ring if and only if Z/2Z is not a homomorphic image of R.