Flat Module

A flat module is a left unital R-module over a unit ring for which the tensor product functor is an exact functor on right modules. Over a commutative ring , this is equivalent to exactness of the tensor product functor .

Every module over satisfies

but the implications cannot in general be reversed.

A module over is flat iff its underlying Abelian group is a torsion-free group. Hence, the rational numbers and the infinite group direct product are flat modules over , but they are not projective. Over a commutative ring that is Noetherian or local, every finitely generated module that is flat is projective (Stacks Project Authors 2026). This property, together with Serre's problem, implies that the three properties above are equivalent for a finitely generated module over a polynomial ring , where is a field.