Remark

If $E$ is a faithfully flat $A$-module, $E$ is a faithful $A$-module: for, if an element $a \in A$ is such that $xa = 0$ for all $x \in E$, the homothety $h : b \mapsto ba$ in $A$ is such that $1, \otimes h = 0$; whence $h = 0$ by property (c) of Proposition 1, that is $a = 0$ since $A$ has a unit element.

Examples (1) The direct sum of a flat module and faithfully flat module is a faithfully flat module by virtue of property (d) of Proposition 1 and § 2, no. 3, Proposition 2. (2) As $A$, is faithfully flat by virtue of criterion (d) of Proposition 1 and § 2, no. 4, Example 1, it follows from (1) that every free module not reduced to 0 is faithfully flat. On the other hand, there exist non-zero direct factors of free modules (in other words, non-zero projective modules) which are faithful but not faithfully flat (Exercise 2). (3) Let $A$ be a principal ideal domain. For an $A$-module $E$ to be faithfully flat, it is necessary and sufficient that it be torsion-free and $E \neq Ep$ for every irreducible element (Algebra, Chapter VII, § 1, no. 3) $p$ of $A$; this follows immediately from § 2, no. 4, Proposition 3 and criterion (d) of Proposition 1. (4) Example (3) shows that the $\mathbf{Z}$-module $\mathbf{Q}$ is a flat and faithful module, but not faithfully flat.