Quasi-Normal Rings

Abstract

A ring R is defined to be quasi-normal if ae = 0 implies eaRe = 0 for a ∈ N(R) and e ∈ E(R), where E(R) and N(R) stand, respectively, for the set of idempotents and the set of nilpotents of R. It is proved that R is quasi-normal if and only if eR(1 − e)Re = 0 for each e ∈ E(R) if and only if T n (R, R) is quasi-normal for any positive integer n. And it follows that for a quasi-normal ring R, R is Π-regular if and only if N(R) is an ideal of R and R/N(R) is regular. Also, using quasi-normal ring, we proved the following: (1) R is an abelian ring if and only if R is a quasi-normal left idempotent reflexive ring; (2) R is a strongly regular ring if and only if R is a von Neumann regular quasi-normal ring; (3) Let R be a quasi-normal ring. Then R is a clean ring if and only if R is an exchange ring; (4) Let R be a quasi-normal Π-regular ring. Then R is a (S,2)-ring if and only if ℤ/2ℤ is not a homomorphic image of R.