Integral extension implies surjective map on spectra

Statement

Suppose $S$ is an integral extension of a ring $R$, in other words $f:R\to S$ is an injective homomorphism of commutative unital rings with the property that every element of $S$ is integral over the image of $R$. Then, the map:

$f^{*}:Spec(S)\to Spec(R)$

from the spectrum of $S$ to that of $R$, that sends a prime ideal of $S$ to its contraction in $R$, is surjective. In other words, every prime ideal of $R$ occurs as the contraction of a prime ideal of $S$.

Proof

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