Bulletin of the
Korean Mathematical Society BKMS

In this paper, we establish necessary and sufficient conditions for an exchange ring $R$ to satisfy the $n$-stable range condition. It is shown that an exchange ring $R$ satisfies the $n$-stable range condition if and only if for any regular $a\in R^n$, there exists a unimodular $u\in ^nR$ such that $au\in R$ is a group member, and if and only if whenever $a\overline{\sim}_nb$ with $a\in R, b\in \mbox{M}_n(R)$, there exist $u\in R^n, v\in ^nR$ such that $a=ubv$ with $uv=1$. As an application, we observe that exchange rings satisfying the $n$-stable range condition can be characterized by Drazin inverses. These also give nontrivial generalizations of [7, Theorem 10], [13, Theorem 10], [15, Theorem] and [16, Theorem 2A].

Keywords: exchange ring, stable range condition, $n$-pseudo similarity

MSC numbers: 16B10, 16S99, 16E50