Induced map on spectra by a ring homomorphism

Suppose $f:R\to S$ is a homomorphism of commutative unital rings. Then, if $Spec(R)$ and $Spec(S)$ denote the spectra of the rings $R$ and $S$ respectively, we have a functorially induced mapping:

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

sending a prime ideal of $S$ to its contraction in $R$. The well-definedness of this map rests on the fact that the contraction of a prime ideal is prime.

Various ring-theoretic assumptions on the nature of $f$ give set-theoretic or topological information about the map $f^{*}$.

Surjective homomorphism yields injective map

If $f:R\to S$ is surjective, the map $f^{*}:Spec(S)\to Spec(R)$ is injective. In fact, $Spec(S)$ is identified with those elements of $Spec(R)$ that contain the kernel of $f$.

Thus, the map is injective and its image is a closed subset. In fact, the map is a closed map with respect to the topologies on both sides.

This can be better understood in relation with the Galois correspondence between a ring and its spectrum.

Localization yields injective map

If $S=U^{-1}R$ for some multiplicative subset $U$ of $R$, and $f$ is the natural map from $R$ to $S$, then $f^{*}$ is again an injective map, but this time the image is not necessarily a closed set. $f^{*}$ identifies $Spec(S)$ with those prime ideals of $R$ that do not intersect $U$.