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