Galois correspondence of extension and contraction

Statement

Suppose $f:R\to S$ is a homomorphism of commutative unital rings. Then, let $Ideals(R)$ and $Ideals(S)$ 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 $S$:

$\_^{e}:Ideals(R)\to Ideals(S)$

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

$\_^{c}:Ideals(S)\to Ideals(R)$