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.

Key Words:

• Abelian rings
• Quasi-normal rings
• Π-Regular rings
• (S, 2) rings

2000 Mathematics Subject Classification:

• 16A30
• 16A50
• 16E50
• 16D30

ACKNOWLEDGMENTS

We would like to thank the referee for his/her helpful suggestions and comments. This project is supported by the Foundation of Natural Science of China (10771182) and the Scientific Research Foundation of Graduate School of Jiangsu Province (CX09B–309Z)

Notes

Communicated by J. P. Zhang.