Exchange rings satisfying stable range conditions

Abstract

In this paper, we establish necessary andsufficient conditions for an exchange ring R to satisfy then-stable range condition. It is shown that an exchange ring Rsatisfies the n-stable range condition if and only if for anyregular ain R^n, there exists a unimodular uin ^nR such thatauin R is a group member, and if and only if wheneveraoverline{sim}_nb with ain R, bin mbox{M}_n(R), thereexist uin R^n, vin ^nR such that a=ubv with uv=1. As anapplication, we observe that exchange rings satisfying then-stable range condition can be characterized by Drazininverses. These also give nontrivial generalizations of [7,Theorem 10], [13, Theorem 10], [15, Theorem] and [16, Theorem 2A].