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

Linear functionals preserving the units of a Riesz space