Proposition 13

Let $A, A'$ be two rings, $X = \mathrm{Spec}(A), X' = \mathrm{Spec}(A')$ and $h$ a homomorphism from $A$ to $A'$; the mapping $^a h : p' \mapsto \overline{h}(p')$ from $X'$ to $X$ is continuous.

For $M \subset A$, the set $(^a h)^{-1}(V(M))$ is the set of prime ideals $p'$ of $A'$ such that $M \subset \overline{h}(p')$, which is equivalent to $h(M) \subset p'$; this set is then equal to $V(h(M))$ and is therefore closed.

We call $^a h$ the mapping associated with the homomorphism $h$.