ON ALEPH(0)-QUASI-CONTINUOUS EXCHANGE RINGS

An associative ring R with identity is said to have stable range one i f for any a,b is an element of R with aR + bR = R, there exists y is a n element of R such that a + by is left (equivalently, right) invertib le. The main results of this note are Theorem 2: A left or right conti nuous ring R has stable range one if and only if R is directly finite (i.e. xy = 1 implies yz = 1 for all x, y is an element of R), Theorem 6: A left or right N-0-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or rig ht N-0-quasi-continuous strongly pi-regular rings have stable range on e. Theorem 6 generalizes a well-known result of Goodearl [10], which s ays that a directly finite, right N-0-continuous von Neumann regular r ing is unit-regular.