  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.
