Flat module

A module whose tensor product functor preserves exactness.

Flat module

A flat module is a left $R$ -module $M$ over a ring $R$ such that tensoring with $M$ preserves exactness: for every short exact sequence of right $R$ -modules

0\to A\to B\to C\to 0,

the sequence

0\to A\otimes_R M\to B\otimes_R M\to C\otimes_R M\to 0

is exact, where $\otimes_R$ is the tensor product . Equivalently, the functor $-\otimes_R M$ is exact (it is always right-exact, so flatness is precisely left-exactness).

Flatness is weaker than projectivity: every projective module is flat (see projective implies flat ), and every free module is flat. Over a commutative ring , flatness controls base change and localization.

Examples:

Any free $R$ -module is flat.
If $R$ is commutative and $S\subseteq R$ is multiplicative, then the localization $S^{-1}R$ is flat as an $R$ -module.
Over a PID , every torsion-free module is flat.