Let be a commutative ring with unity. A free module over is a (unital) module isomorphic to a direct sum of copies of . In particular, as every abelian group is a -module, a free abelian group is a direct sum of copies of . This is equivalent to saying that the module has a free basis, i.e. a set of elements with the property that every element of the module can be uniquely expressed as an linear combination over of elements of the free basis. In the case that a free module over is a sum of finitely many copies of , then the number of copies is called the rank of the free module.

An alternative definition of a free module is via its universal property: Given a set , the free -module on the set is equipped with a function satisfying the property that for any other -module and any function , there exists a unique -module map such that .