Geodesic
Clean matrices over commutative rings
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 145-158

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A matrix $A\in M_n(R)$ is $e$-clean provided there exists an idempotent $E\in M_n(R)$ such that $A-E\in \mathop{\rm GL}_n(R)$ and $\det E=e$. We get a general criterion of $e$-cleanness for the matrix $[[a_1,a_2,\cdots ,a_{n+1}]]$. Under the $n$-stable range condition, it is shown that $[[a_1,a_2,\cdots ,a_{n+1}]]$ is $0$-clean iff $(a_1,a_2,\cdots ,a_{n+1})=1$. As an application, we prove that the $0$-cleanness and unit-regularity for such $n\times n$ matrix over a Dedekind domain coincide for all $n\geq 3$. The analogous for $(s,2)$ property is also obtained.
A matrix $A\in M_n(R)$ is $e$-clean provided there exists an idempotent $E\in M_n(R)$ such that $A-E\in \mathop{\rm GL}_n(R)$ and $\det E=e$. We get a general criterion of $e$-cleanness for the matrix $[[a_1,a_2,\cdots ,a_{n+1}]]$. Under the $n$-stable range condition, it is shown that $[[a_1,a_2,\cdots ,a_{n+1}]]$ is $0$-clean iff $(a_1,a_2,\cdots ,a_{n+1})=1$. As an application, we prove that the $0$-cleanness and unit-regularity for such $n\times n$ matrix over a Dedekind domain coincide for all $n\geq 3$. The analogous for $(s,2)$ property is also obtained.
Classification : 15A23, 16E50
Keywords: matrix; clean element; unit-regularity 
Chen, Huanyin. Clean matrices over commutative rings. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 145-158. http://geodesic.mathdoc.fr/item/CMJ_2009_59_1_a9/
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