STABLE RANGE ONE FOR RINGS WITH MANY IDEMPOTENTS

An associative ring R is said to have stable range 1 if for any a, b is an element of R satisfying aR + bR = R, there exists y is an element of R such that a + by is a unit. The purpose of this note is to prove the following facts. Theorem 3: An exchange ring R has stable range 1 if and only if every regular element of R is unit-regular. Theorem 5: If R is a strongly pi-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menal [5].