Let $A$ be a preordered ring (or $\mathbb{R}$-algebra or $\mathbb{Q}$-algebra).
Say that ${a \in A}$ is totally positive if for every morphism ${f : A \rightarrow \mathbb{R}}$ of ordered algebras, ${f a \geq 0}$.
Define totally negativeness analogously.
To avoid trival cases assume that there is some morphism (of preordered rings) ${A \rightarrow \mathbb{R}}$.
Assume also that $A$ with bounded inversion (i.e., for any positive ${a \in A}$, ${1 + a}$ is a unit).
Question: Is it the case that any unit of $A$ is totally positive or totally negative?
Thanks.