Exercise 3, § 5

Let $A$ be a ring and $P$ a finitely generated projective $A$-module. Show that $A$ is isomorphic to a finite product of rings $\prod_{i \in I} A$, and $P$ is isomorphic to a product $\prod_{i \in I} P_i$, where $P_i$ is a projective $A_i$-module of rank $n_i$, the $n_i$ being distinct (use Theorem 1 and the fact that $\mathrm{Spec}(A)$ is quasi-compact).