On Rings Whose Primitive Factor Rings Are Left Artinian

Abstract

In this paper, we prove that if R is a Min-E ring, then the following statements are equivalent: (1) Every left primitive factor ring of R is left artinian; (2) R is a π-regular ring; (3) R is an Exchange ring; (4) R is a Clean ring. As an application, we obtain that if R is Min-E ring, every left primitive factor ring of R is artinian and any direct product of them is a Min-E ring, then every non-zero homomorphic image of R contains only a finite number of prime ideals. When R is Min-E ring and has no infinite set of orthogonal idempotents, a left R-module M is Min-E if and only if M is Max-E. Also, we show, if R is a strongly clean Min-E ring, then R is directly finite and has stable range 1.