Exchange Rings Satisfying the n-Stable Range Condition, II

Abstract

This is a continuation of the paper [14]. It is shown that any finite subdirect product of exchange rings satisfying the n-stable range condition is still an exchange ring satisfying the n-stable range condition. Furthermore, we give necessary and sufficient conditions on matrices over an exchange ring R, under which R satisfies the n-stable range condition. This generalizes the corresponding results for unit-regular rings and the stable range one condition.