Strongly 𝜋-regular rings have stable range one

Abstract

A ring R R is said to be strongly π \pi -regular if for every a ∈ R a\in R there exist a positive integer n n and b ∈ R b\in R such that a n = a n + 1 b a^{n}=a^{n+1}b . For example, all algebraic algebras over a field are strongly π \pi -regular. We prove that every strongly π \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 M cancels from direct sums whenever End R ( M ) \text {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.