# Turkish Journal of Mathematics

## DOI

10.3906/mat-1007-364

## Abstract

A ring R is defined to be weakly normal if for all a, r \in R and e \in E(R), ae = 0 implies Rera is a nil left ideal of R, where E(R) stands for the set of all idempotent elements of R. It is proved that R is weakly normal if and only if Rer(1-e) is a nil left ideal of R for each e \in E(R) and r \in R if and only if T_n(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.

## Keywords

Weakly normal rings, Abelian rings, regular rings, quasi-normal rings, semiabelian rings, exchange rings, clean rings

## First Page

47

## Last Page

57

## Recommended Citation

WEİ, JUNCHAO and LI, LIBIN
(2012)
"Weakly normal rings,"
*Turkish Journal of Mathematics*: Vol. 36:
No.
1, Article 4.
https://doi.org/10.3906/mat-1007-364

Available at:
https://journals.tubitak.gov.tr/math/vol36/iss1/4