Journal of the
Korean Mathematical Society JKMS

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 $u,v\in U(R)$ if and only if for any $a\in R$, there exist $u,v\in U(R)$ such that $a=u+v$. Furthermore, there exists $u\in U(R)$ such that $1_R\pm u\in U(R)$ if and only if for any $a\in R$, there exists $u\in U(R)$ such that $a\pm u\in U(R)$. We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.

Keywords: exchange ring, Pierce stalk, stable ring

MSC numbers: 16E50, 19U99