Weakly normal rings
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 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.
Anahtar Kelimeler:
Weakly normal rings, Abelian rings, regular rings, quasi-normal rings, semiabelian rings, exchange rings, clean rings
Weakly normal rings
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 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.
Keywords:
Weakly normal rings, Abelian rings, regular rings, quasi-normal rings, semiabelian rings, exchange rings, clean rings,
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- Junchao WEI, Libin LI School of Mathematics, Yangzhou University, Yangzhou, 225002, P. R. CHINA e-mail: jcweiyz@yahoo.com.cn
- ISSN: 1300-0098
- Yayın Aralığı: Yılda 6 Sayı
- Yayıncı: TÜBİTAK
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