Clean matrices over commutative rings

Chen, Huanyin. "Clean matrices over commutative rings." Czechoslovak Mathematical Journal 59.1 (2009): 145-158. <http://eudml.org/doc/37913>.

@article{Chen2009,
abstract = {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 $n\ge 3$. The analogous for $(s,2)$ property is also obtained.},
author = {Chen, Huanyin},
journal = {Czechoslovak Mathematical Journal},
keywords = {matrix; clean element; unit-regularity; matrix; clean element; unit-regularity; idempotent; Dedekind domain},
language = {eng},
number = {1},
pages = {145-158},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Clean matrices over commutative rings},
url = {http://eudml.org/doc/37913},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Chen, Huanyin
TI - Clean matrices over commutative rings
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 1
SP - 145
EP - 158
AB - 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 $n\ge 3$. The analogous for $(s,2)$ property is also obtained.
LA - eng
KW - matrix; clean element; unit-regularity; matrix; clean element; unit-regularity; idempotent; Dedekind domain
UR - http://eudml.org/doc/37913
ER -