Flat module

Remarks

Since $- \otimes_R M$ is always right exact, $M$ is flat exactly when $- \otimes_R M$ takes exact sequences to exact sequences. It is enough to check the condition for the inclusions $I \to R$ of finitely generated ideals: $M$ is flat if and only if $I \otimes_R M \to M$ is injective for each such $I$. Over a principal ideal domain, a module is flat exactly when it is torsion-free. Projective modules are flat, and a finitely presented flat module is projective.

Example: $\Q$ is a flat $\Z$-module. For any $\Z$-module $N$, the module $N \otimes_\Z \Q$ is the localization $S^{-1}N$ with $S = \Z \setminus \set{0}$, and localization preserves injective maps.

Non-example: $\Z/2\Z$ is not a flat $\Z$-module. The map $f \colon \Z \to \Z$, $x \mapsto 2x$, is injective, but $f \otimes \id$ is multiplication by $2$ on $\Z \otimes_\Z \Z/2\Z \cong \Z/2\Z$, which is the zero map on a nonzero module.