Faithfully flat module

Remarks

The following are equivalent for an $R$-module $M$:

1. $M$ is faithfully flat.
2. A sequence $N' \to N \to N''$ of $R$-modules is exact if and only if $N' \otimes_R M \to N \otimes_R M \to N'' \otimes_R M$ is exact.
3. $M$ is flat and $\mathfrak{m} M \ne M$ for every maximal ideal $\mathfrak{m}$ of $R$.

Example: a nonzero free module $M = R^{(I)}$ is faithfully flat. It is flat, and $N \otimes_R R^{(I)} \cong N^{(I)}$, which is zero only when $N = 0$, since $I$ is not empty.

Non-example: $\Q$ is a flat $\Z$-module, so clause 1 holds. But $\Z/2\Z \otimes_\Z \Q = 0$ while $\Z/2\Z \ne 0$, since $1 \otimes q = 2 \otimes \frac{q}{2} = 0 \otimes \frac{q}{2} = 0$; so clause 2 fails.