Strongly $\pi$-regular rings have stable range one

Abstract:

A ring $R$ is said to be strongly $\pi$-regular if for every $a\in R$ there exist a positive integer $n$ and $b\in R$ such that $a^{n}=a^{n+1}b$. For example, all algebraic algebras over a field are strongly $\pi$-regular. We 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 $\text {End}_{R} (M)$ has stable range one. As a consequence of our main result and Evans’ Theorem, modules satisfying Fitting’s Lemma cancel from direct sums.