Weakly normal rings

Authors : Junchao WEI, Libin LI

Pages : 47-57

View : 15 | Download : 5

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :A ring R is defined to be weakly normal if for all a, r \in R and e \in Einsert ignore into journalissuearticles values(R);, ae = 0 implies Rera is a nil left ideal of R, where Einsert ignore into journalissuearticles values(R); stands for the set of all idempotent elements of R. It is proved that R is weakly normal if and only if Rerinsert ignore into journalissuearticles values(1-e); is a nil left ideal of R for each e \in Einsert ignore into journalissuearticles values(R); and r \in R if and only if Tninsert ignore into journalissuearticles values(R, R); is weakly normal for any positive integer n. And it follows that for a weakly normal ring R insert ignore into journalissuearticles values(1); R is Abelian if and only if R is strongly left idempotent reflexive; insert ignore into journalissuearticles values(2); R is reduced if and only if R is n-regular; insert ignore into journalissuearticles values(3); R is strongly regular if and only if R is regular; insert ignore into journalissuearticles values(4); R is clean if and only if R is exchange. insert ignore into journalissuearticles values(5); exchange rings have stable range 1.
Keywords : Weakly normal rings, Abelian rings, regular rings, quasi normal rings, semiabelian rings, exchange rings, clean rings