Unit-Regularity of Regular Nilpotent Elements

Abstract

Let $a$ be a regular element of a ring $R$. If either $K:={\mathrm{r}}_{\mathrm{R}}(\mathrm{a})$ has the exchange property or every power of $a$ is regular, then we prove that for every positive integer $n$ there exist decompositions $$R_R=K\oplus X_n\oplus Y_n=E_n\oplus X_n\oplus a{Y}_n,$$ where $Y_n\subseteq a^{n}R$ and $E_n\cong R/aR$. As applications we get easier proofs of the results that a strongly $\pi$-regular ring has stable range one and also that a strongly $\pi$-regular element whose every power is regular is unit-regular.

In the revision some typos are corrected, minor modifications are made and also references to two related papers are added. To appear in Algebras and Representation Theory