Exercise 8, § 2

Let $A$ be a ring, $(S_\lambda)_{\lambda \in L}$ a family of multiplicative subsets of $A$ and $M = \bigoplus_{\lambda \in L} S_\lambda^{-1}A$. Show that the two following properties are equivalent: ($a$)$M$ is a faithfully flat $A$-module; ($\beta$) for every maximal ideal $m$ of $A$, there exists $\lambda \in L$ such that $m \cap S_\lambda = \varnothing$. In particular, if $S$ is a multiplicative subset of $A$, the $A$-module $S^{-1}A$ is faithfully flat only if the elements of $S$ are invertible, in which case $S^{-1}A$ is identified with $A$.