Exchange Rings with Artinian Primitive Factors

Abstract

We prove that every exchange ring with primitive factors Artinian is clean. Also, it is shown that for exchange rings with Artinian primitive factors, the following are equivalent: (1) Every element in R is a sum of two units. (2) There exist α, β ∈ U(R) such that α + β = 1. (3) R does not have Z / 2 Z as a homomorphic image. Finally, we prove that exchange ring R is strongly π-regular if the Jacobson radical of any homomorphic image of R is T-nilpotent or locally nilpotent. These are generalizations of the corresponding results of A. Badawi, W. D. Burgess and P. Menal, Fisher and Snider, and J. Stock.