Pierce Stalks of Exchange Rings

Abstract

Abstract. We prove, in this article, that a ring R is a stable exchange ring if and only if so are all its Pierce stalks. If every Pierce stalks of R is artinian, then 1 R = u + v with u, v ∈ U (R) if and only if for any a ∈ R, there exist u, v ∈ U (R) such that a = u + v. Furthermore, there exists u ∈ U (R) such that 1 R ± u ∈ U (R) if and only if for any a ∈ R, there exists u ∈ U (R) such that a ± u ∈ U (R). We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.