Geodesic
Strong separativity over exchange rings
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 417-428 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A \oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\begin{pmatrix} a\\ ** \end{pmatrix} \in M_2(S)$. The dual assertions are also proved.
An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A \oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\begin{pmatrix} a\\ ** \end{pmatrix} \in M_2(S)$. The dual assertions are also proved.
Classification : 16D70, 16E50, 19B10, 19E99
Keywords: strong separativity; exchange ring; regular ring 
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     author = {Chen, Huanyin},
     title = {Strong separativity over exchange rings},
     journal = {Czechoslovak Mathematical Journal},
     pages = {417--428},
     year = {2008},
     volume = {58},
     number = {2},
     mrnumber = {2411098},
     zbl = {1166.16002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a7/}
}
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Chen, Huanyin. Strong separativity over exchange rings. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 2, pp. 417-428. http://geodesic.mathdoc.fr/item/CMJ_2008_58_2_a7/

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