Linear functionals preserving the units of a Riesz space

Let $L$ be a Riesz space with a strong unit $e>0$.$$We show that a unital linear functional $H:L\to \mathbb{R}$ satisfies $% H\left( u\right) \neq 0$foranystrongunit$u\in L$ifandonlyif$H$ acts like a Riesz homomorphism on every $e$-clean vector subspace of $L\mathrm{.}$ We deduce that $L$ is $e$-clean if and only if any unital linear functional $% H:L\rightarrow $\mathbb{R}$$suchthat$H\left( u\right) \neq 0$foranystrongunit$u\in L$ is a Riesz homomorphism.