On N0-quasi-continuous exchange rings

Abstract

An associative ring R with identity is said to have stable range one if for any a,b∊ R with aR + bR = R, there exists y ∊ R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ∊ R), Theorem 6: A left or right N 0o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N 0-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N o-continuous von Neumann regular ring is unit-regular