ring homomorphism and spectrum

LEMMA 1.5   Let $A$ ,$B$ be two ring homomorphisms. Let

$$\alpha: A\to B$$

be a ring homomorphism (which we always assume to be unital).

Then we have a associate map

$$\operatorname{Spec}(\alpha): \operatorname{Spec}(B)\to \operatorname{Spec}(A)$$

defined by

$$\operatorname{Spec}(\alpha)(\mathfrak{p})=\alpha^{-1}(\mathfrak{p}) \qquad(\forall \mathfrak{p}\in \operatorname{Spec}(B)).$$

The map $\operatorname{Spec}(\alpha)$ has the following properties.

1. $$\operatorname{Spec}(\alpha)(\mathfrak{p})=\{ f \in A; \rho_\mathfrak{p}(\alpha(f))=0 \}$$

2. $$\operatorname{Spec}(\alpha)^{-1}(O_f)=O_{\alpha(f)}$$
   for any $f\in A$ .
3. $\operatorname{Spec}(\alpha)$ is continuous.