Exercise 5, § 3

Let $A$ be a commutative ring and $M$ a flat $A$-module containing a submodule $N$ which is not a flat module (cf. $§ 2$, Exercise 3). Let $B$ (resp. $C$) be the $A$-module $A \oplus N$ (resp. $A \oplus M$) in which multiplication is defined by $(a, x)(a', x') = (aa', ax' + a'x)$; then $B$ is not a flat $A$-module, but the $B$-module $C$ is a faithfully flat $A$-module and consequently $B$ satisfies the conditions of Proposition 8 of no. 5.