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.
In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n(I)$, there exist left invertible $U_1,U_2\in M_n(R)$ and right invertible $V_1,V_2\in M_n(R)$ such that $U_1V_1AU_2V_2= \operatorname{diag}(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.
Let $R$ be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in $R$ is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.
We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E(R)$ and a $u\in U(R)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann(a^+)$ and $u\in U(R)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR\Longrightarrow aR\cong bR$.
It is shown that a ring $R$ is a $GM$-ring if and only if there exists a complete orthogonal set $\lbrace e_1,\cdots ,e_n\rbrace $ of idempotents such that all $e_iRe_i$ are $GM$-rings. We also investigate $GM$-rings for Morita contexts, module extensions and power series rings.
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 $\begin{pmatrix} a&b\\ *&* \end{pmatrix} \in M_2(S)$. The dual assertions are also proved.