Galois correspondence of extension and contraction

Statement

Suppose $f : R \to S$ is a homomorphism of commutative unital rings. Then, let $I d e a l s (R)$ and $I d e a l s (S)$ denote the set of ideals in $R$ and $S$ respectively, viewed as partially ordered sets by inclusion. Consider the following maps:

The extension map which sends an ideal of $R$ to the ideal generated by its image in $S$ :

$_^{e} : I d e a l s (R) \to I d e a l s (S)$

The contraction map which sends an ideal of $S$ to its full inverse image in $R$ :

$_^{c} : I d e a l s (S) \to I d e a l s (R)$