Effect of ideal contraction on Galois correspondent

This fact is an application of the following pivotal fact/result/idea: nilradical of subring lemma
View other applications of nilradical of subring lemma OR Read a survey article on applying nilradical of subring lemma

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.