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 $$R$$:

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