Weakly normal rings

Ar ingR is defined to be weakly normal if for all a, r ∈ R and e ∈ E(R ), ae = 0 implies Rera is a nil left ideal of R ,w hereE(R) stands for the set of all idempotent elements of R .I t is proved thatR is weakly normal if and only if Rer(1 − e) is a nil left ideal of R for each e ∈ E(R )a ndr ∈ R if and only if Tn(R, R) is weakly normal for any positive integer n . And it follows that for a weakly normal ring R (1) R is Abelian if and only if R is strongly left idempotent reflexive; (2) R is reduced if and only if R is n -regular; (3) R is strongly regular if and only if R is regular; (4) R is clean if and only if R is exchange. (5) exchange rings have stable range 1.