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Keywords:
exchange ring; stable range one; idempotent; unit
Summary:
We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E(R)$ and a $u\in U(R)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann(a^+)$ and $u\in U(R)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR\Longrightarrow aR\cong bR$.
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