Strongly π-regular rings have stable range one

A ring R is said to be strongly π-regular if for every a ∈ R there exist a positive integer n and b ∈ R such that an = an+1b. For example, all algebraic algebras over a field are strongly π-regular. We prove that every strongly π-regular ring has stable range one. The stable range one condition is especially interesting because of Evans' Theorem, which states that a module AI cancels from direct sums whenever EndR(M) has stable range one. As a consequence of our main result and Evans' Theorem, modules satisfying Fitting's Lemma cancel from direct sums. ©1996 American Mathematical Society.