UNITS, IDEMPOTENTS, AND STABLE RANGE CONDITIONS

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).

Keywords:

• Stable range conditions
• Matrix ring

ACKNOWLEDGMENTS

The author was supported by National Natural Science Foundation of China, Grant No. 19801012, and the Ministry of Education of China.