Exercise 20, § 4

Give an example of a ring homomorphism $\phi : A \to B$ such that every maximal ideal of $A$ is of the form $\phi(m)$, where $m$ is a maximal ideal of $B$, but there exist prime ideals of $A$ which are not inverse images under $\phi$ of ideals of $B$ (take $B$ to be a field).