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Every projective module is flat, so tensoring with it preserves exact sequences.
Let be a projective right -module. Then is flat: for every short exact sequence of left -modules
the induced sequence
is exact.
Projective modules are direct summands of free modules. The functor is therefore a direct summand of a direct sum of identity functors, so it preserves injections; as every tensor functor is right exact, it is exact.
An -module is projective if for every surjective homomorphism and every homomorphism (see module homomorphisms), there exists a homomorphism such that .
A flat module is a left -module over a ring such that tensoring with preserves exactness: for every short exact sequence of right -modules
the sequence
is exact, where is the tensor product.
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.
A left -module is free if it has a basis , i.e. every element of admits a unique expression as a finite -linear combination of elements of .
Free modules are characterized by the universal property of free modules: functions from a basis extend uniquely to module homomorphisms. Over a field, free modules coincide with vector spaces and their bases agree with linear-algebra bases.