Exercise 10, § 1

(a) Now prove that a faithfully flat module is faithful (II, p. 28). (b) The $\mathbf{Z}$-module Q is faithful and flat, but not faithfully flat. (c) Let $(p_n)$ be the strictly increasing sequence of prime numbers, and let A be the product ring $\prod_n \mathbf{Z}/p_n \mathbf{Z}$. Now prove that the direct sum E of the $\mathbf{Z}/p_n \mathbf{Z}$ is a projective faithful a-module which is not faithfully flat (observe that E is an ideal of A such that $E^2 = E$).