I'm attempting Atiyah-Macdonalds Exercise 5.1:
Let $f: A \to B$ be an integral homomorphism of rings. Show that $f^{\ast}: \text{Spec}(B) \to \text{Spec}(A)$ is a closed mapping, i.e. that it maps closed sets to closed sets. (This is a geometrical equivalent of (5.10)).
Here $f$ being integral means that $B$ is integral over $f(A)$. We want to show that for any closed subset $C \subset \text{Spec}(B)$, $f^{\ast}(C) = \{f^{-1}(\mathfrak{p}) | \mathfrak{p}) \in C \}$ is closed.
I'm quite stuck at the beginning, in particular I'm not sure how to make connections between integral and the topology on spectrum of an ring. Any help is appreciated!