Strongly pi-regular rings have stable range one

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