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.
@article{CMJ_2009__59_1_a9,
author = {Chen, Huanyin},
title = {Clean matrices over commutative rings},
journal = {Czechoslovak Mathematical Journal},
pages = {145--158},
publisher = {mathdoc},
volume = {59},
number = {1},
year = {2009},
mrnumber = {2486621},
zbl = {1224.15034},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMJ_2009__59_1_a9/}
}
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/