Exercise 2, § 3

Let $(p_n)$ be the strictly increasing sequence of prime numbers and $A$ the product ring $\prod_n \mathbf{Z}/p_n \mathbf{Z}$. Show that the direct sum $E$ of the $\mathbf{Z}/p_n \mathbf{Z}$ is a faithful projective $A$-module which is not faithfully flat (observethat $E$ is an ideal of $A$ such that $E^2 = E$).