Exercise 1, § 2

Give an example of an injective ring homomorphism $\rho : A \to B$ such that B is finite A-algebra, the mapping $^a\rho : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$ is surjective but B is not a flat A-module. Conversely, give an example of a faithfully flat A-algebra which is finitely generated but not integral over A.