Some notes on nil-semicommutative rings

Abstract

A ring R is dened to be nil-semicommutative if ab 2 N(R) implies arb 2 N(R) for a;b;r 2 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.