Exercise 19, § 4

(a) Let $\phi : A \to B$ be a ring homomorphism. Show that, for every ideal $b$ of $B$, $\overline{\phi(V(b))} = V(\phi^{-1}(b))$.

(b) Deduce from (a) that, for $^a\phi(\operatorname{Spec}(B))$ to be dense in $\operatorname{Spec}(A)$, it is necessary and sufficient that $\operatorname{Ker}(\phi)$ be a nil ideal of $A$.

(c) Show that, if every $b \in B$ may be written as $b = h\phi(a)$, where $h$ is invertible in $B$ and $a \in A$, $^a\phi$ is a homeomorphism of $\operatorname{Spec}(B)$ onto a subspace of $\operatorname{Spec}(A)$.