Abstract
In this paper, we show that if R satisfies unit 1-stable range, then so does M n (R) for any n ≥ 1. Also we prove that if R has many unit-regular elements, then so does M n (R) for any n ≥ 1. Finally, we show that if for any x, y ∈ R there exists a u ∈ U (R) such that x−u, y−u −1 ∈ U (R), then for any X, Y ∈ M n (R), there exists a U ∈ GL n (R) such that X–U, Y−U −1 ∈ GL n (R).
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ACKNOWLEDGMENTS
The author was supported by National Natural Science Foundation of China, Grant No. 19801012, and the Ministry of Education of China.