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An exchange ring $R$ is separative provided that for all finitely generatedprojective right $R$-modules $A$ and $B$, $A\oplus A\cong A\oplus B\congB\oplus B\Longrightarrow A\cong B$. Let $R$ be a separative exchange ring inwhich $2$ is invertible, and let $a-a^3\in R$ be regular. We prove, in thisnote, that $a\in R$ is unit-regular if $R(1-a^2)R=Rr(a)={\ell}(a)$. An element$a$ in a ring $R$ is special clean if there exists an idempotent $e\in R$ suchthat $a-e\in R$ is a unit and $aR\bigcap eR=0$. Furthermore, we prove that$a\in R$ is special clean if $aR/ar(a^2), R/\big(aR+r(a)\big)$ are projective,and $R(a-a^3)R=Rar(a^2)=\ell (a^2)aR$. These also extend the correspondingresults in separative regular rings.

Separative exchange rings in which 2 is invertible