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Czechoslovak Mathematical Journal, Vol. 57, No. 2, pp. 579-590, 2007
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Exchange rings with stable range one

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Huanyin Chen

Department of Mathematics, Hunan Normal University, Changsha, 410006, People's Republic of China, e-mail: ` chyzxl@hunnu.edu.cn`

**Abstract:** 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 $a,b\in R$, $R/aR\cong R/bR\Longrightarrow aR\cong bR$.

**Keywords:** exchange ring, stable range one, idempotent, unit

**Classification (MSC 2000):** 16E50, 16U99

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