Effect of ideal contraction on Galois correspondent

Statement
Suppose $$f:R \to S$$ is a homomorphism of commutative unital rings, and $$I$$ is an ideal of $$S$$. Suppose $$\mathcal{Z}(I)$$ denotes the subset of $$Spec(S)$$ comprising the prime ideals which contain $$I$$ (the Galois correspondent to $$I$$ under the Galois correspondence between a ring and its spectrum). Then:

$$\mathcal{Z}(I^c) = \overline{f^*\mathcal{Z}(I))}$$

An analogous statement is true for the max-spectrum, if we assume that both $$R$$ and $$S$$ are Jacobson rings. This is to equate the Jacobson radical with the nilradical for every quotient ring.