Stable range one for rings with many idempotents

Abstract:

An associative ring $R$ is said to have stable range $1$ if for any $a$, $b \in R$ satisfying $aR + bR = R$, there exists $y \in R$ such that $a + by$ 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 [$5$].