Rings All of Whose Left ideals are Generated by Idempotents

Abstract:

      A ring R is called a left PI-ring if every principal left ideal in R is generated by a finite set of idempotents. The aim of this paper is to study von Neumann regularity of left PI-rinss.We prove the following results: (1) A ring R is artinian semisimple if and only if R is an orthogonally finite left PI-ring; (2) A ring R is strongly regular if and only if R is a left PI-ring and R /P is a division ring for any prime ideal P of R: (3) A ring R is regular and allleft primitive factor rings of R are artinian;(4)A ring R is a left self-injective regular ring and soc(_ＲR)≠0 if and only if R is a left PI-ring containing an injective maximal left ideal; (5) A ring R is an MELT regular ring if and only if R is an MELT left PI-ring. We also give some character-izations of normal rings.