Clean matrices over commutative rings

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 th…