PIERCE STALKS OF EXCHANGE RINGS

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 + $\upsilon$ with u, $\upsilon$ $\in$ U(R) if and only if for any a $\in$ R, there exist u, $\upsilon$ $\in$ U(R) such that a = u + $\upsilon$. 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.