Non-invertible elements in an algebra which do not belong to the kernel of any character

Question

Is there a commutative unital algebra $A$ over $\mathbb C$ such that for some $f\in A$, $\chi(f)\not=0$ for every (non-zero) homomorphism $\chi:A\to\mathbb C$, but for which $f$ is not invertible? If $A$ is the algebra of rational functions in $\mathbb C$, then this holds, but in that case there isn't any such character at all. So I ask this question for the case where $X(A)\not=\emptyset$ (the set of all non-zero homomorphisms into $\mathbb C$). The algebra of all holomorphic functions in $\mathbb C$ is not such an example, and of course no Banach-algebra has such a `weird' property.