Proposition 3

Let $(f_i)_{i \in I}$ be a finite family of elements of a ring $A$, generating the ideal $A$ of $A$. The ring $B = \prod_{i \in I} A_i$, is then a faithfully flat $A$*-module*.

By § 2, no. 4, Theorem 1, each of the $A_i$ is a flat $A$-module, hence so is $B$ (Chapter I, § 2, no. 3, Proposition 2). On the other hand, if $p$ is a prime ideal of $A$, there exists an index $i$ such that $f_i \notin p$ and $p_{f_i} = pA_{f_i}$ is therefore a prime ideal of $A_{f_i}$. Then $pB \subset pA_{f_i} \times \prod_{j \neq i} A_j \neq B$ since $pA_{f_i} \neq A_{f_i}$; this suffices to imply that $B$ is a faithfully flat $A$-module (Chapter I, § 3, no. 1, Proposition 1).