Generalized weakly symmetric rings

A ring R is defined to be GWS if abc=0 implies bac⊆N(R) for a,b,c∈R, where N(R) stands for the set of nilpotent elements of R. Since reduced rings and central symmetric rings are GWS, we study sufficient conditions for GWS rings to be reduced and central symmetric. We prove that a ring R is GWS if and only if the n×n upper triangular matrices ring Un(R,R) is GWS for any positive integer n. It is proven that GWS rings are directly finite and left min-abel. For a GWS ring R, R is a strongly regular ring if and only if R is a von Neumann regular ring if and only if R is a left SF ring and J(R)=0; R is an exchange ring if and only if R is a clean ring. Finally, we show that GWS exchange rings have stable range 1 and a GWS semiperiodic ring R with N(R)≠J(R) is commutative.