Exchange Rings Generated by Their Units

Abstract

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈ U(R) such that 1R ± u ∈ U(R), if and only if for any a∈ R, there exists a u ∈ U(R) such that a ± u ∈ U(R). Furthermore, we prove that, for any A ∈ Mn(R)(n ≥ 2), there exists a U ∈ GLn(R) such that A ± U ∈ GLn(R).