Geodesic
Exchange rings with stable range one
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 2, pp. 579-590 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $a,b\in R$, $R/aR\cong R/bR\Longrightarrow aR\cong bR$.
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 $a,b\in R$, $R/aR\cong R/bR\Longrightarrow aR\cong bR$.
Classification : 16D70, 16E20, 16E50, 16U99, 19B10
Keywords: exchange ring; stable range one; idempotent; unit 
@article{CMJ_2007_57_2_a4,
     author = {Chen, Huanyin},
     title = {Exchange rings with stable range one},
     journal = {Czechoslovak Mathematical Journal},
     pages = {579--590},
     year = {2007},
     volume = {57},
     number = {2},
     mrnumber = {2337616},
     zbl = {1161.16008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_2_a4/}
}
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Chen, Huanyin. Exchange rings with stable range one. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 2, pp. 579-590. http://geodesic.mathdoc.fr/item/CMJ_2007_57_2_a4/