Extensions of Rings with Many Units

Abstract

In this paper, we investigate stable range conditions over extensions of matrix rings. It is shown that a ring R satisfies (s, 2)-stable range if and only if R has a complete orthogonal set {e ₁,…, e _n } of idempotents such that all e _i Re _i satisfy (s, 2)-stable range. Also we extend this result to (s, 2)-rings and rings satisfying unit 1-stable range.

Key Words:

• Stable range condition
• Matrix extension
• Idempotent

Acknowledgments

It is a pleasure to thank the referee for excellent suggestions and corrections which simplified Theorem 3.4, Theorem 4.6, and led to the new version. This work was supported by the National Natural Science Foundation of China (Grant No. 19801012) and the Ministry of Education of China ([2000] 65).