flat R-module

Definition

An R-module is flat if the functor from the tensor product

is exact. That is, for all short exact sequences of -modules

the sequence

is exact.

In other words, since by this proposition, the last three nodes are exact, flatness is a condition that for all injective -module homomorphisms , the morphism

is injective.

Examples

• The free -module is flat, since is isomorphic to the identity functor.

• Since for -modules and

  thus, and are flat if and only if is flat. This means that finite rank free R-modules and finitely-generated projective R-modules are flat.