Let $F: \mathbb R^{n} \to \mathbb R^{m} $ be a smooth map, then we have homomorphism of algebras $F^{\*}: C^{\infty}(\mathbb R^{m}) \to C^{\infty}(\mathbb R^{n})$. Is it true that any homomorphism of these algebras $f$ is equal to $F^{\*}$ for some smooth map $F$?

I think that shouldn't be true, because $ \mathbb R^{n}$ isn't compact. But I can't find the example of this homomorphism.