主左理想由若干个幂等元生成的环

Rings All of Whose Left ideals are Generated by Idempotents

环Ｒ称为左ＰＩ－环，是指Ｒ的每个主左理想由有限个幂等元生成．本文的主要目的是研究左ＰＩ－环的ｖｏｎＮｅｕｍａｎｎ正则性，证明了如下主要结果：（１）环Ｒ是Ａｒｔｉｎ半单的当且仅当Ｒ是正交有限的左ＰＩ－环；（２）环Ｒ是强正则的当且仅当Ｒ是左ＰＩ－环，且对于Ｒ的每个素理想Ｐ，Ｒ／Ｐ是除环；（３）环Ｒ是正则的且Ｒ的每个左本原商环是Ａｒｔｉｎ的当且仅当Ｒ是左ＰＩ－环且Ｒ的每个左本原商环是Ａｒｔｉｎ的；（４）环Ｒ是左自内射正则环且Ｓｏｃ（ＲＲ）≠０当且仅当Ｒ是左ＰＩ－环且它包含内射极大左理想；（５）环Ｒ是ＭＥＬＴ正则环当且仅当Ｒ是ＭＥＬＴ左ＰＩ－环．

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 if and only if R is a left PI-ring and all left primitive factor rings are artinian; (d ) 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 ail MELT left PI-ring. We also give some characterisations of normal rings.