Unit-Regularity of Regular Nilpotent Elements

Let $a$ be a regular element of a ring $R$. If either $K:=\rm{r}_R(a)$ has the exchange property or every power of $a$ is regular, then we prove that for every positive integer $n$ there exist decompositions $$ R_R = K \oplus X_n \oplus Y_n = E_n \oplus X_n \oplus aY_n,$$ where $Y_n \subseteq a^nR$ 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.
Comment: 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