科研之友
微信
新浪微博
Facebook
分享链接
摘要
Let R be an exchange ring. In this article, we show that the following conditions are equivalent. (1) R has stable range not more than n; (2) whenever x epsilon R-n is regular, there exists some unimodular regular W epsilon R-n such that x = xwx; (3) whenever X epsilon R-n is regular, there exist some idempotent e epsilon R and some unimodular regular W epsilon R-n such that x = ew; (4) whenever x epsilon R-n is regular, there exist some idempotent e epsilon M-n (R) and some unimodular regular w epsilon R-n such that x = we; (5) whenever a(R-n) + bR = dR with a epsilon R-n and b, d epsilon R, there exist some z epsilon R-n and some unimodular regular w epsilon R-n such that a + bz = dw, (6) whenever x = xyx with x epsilon R-n and y epsilon R-n, there exist some u epsilon R-n and v epsilon R-n such that vxyu = yx and uv = 1. These, by replacing unimodularity with unimodular regularity, generalize the corresponding results of Canfell (1995, Theorem 2.9), Chen (2000, Theorem 4.2 and Proposition 4.6, 2001, Theorem 10), and Wu and Xu (1997, Theorem 9), etc.
- 出版日期2008-7
- 单位南京师范大学
