Let $R$ be a ring. A free module over $R$ is a direct sum of copies of $R$ .

Similarly, as an abelian group is simply a module over $\Bbb{Z}$ , a free abelian group is a direct sum of copies of $\Bbb{Z}$ .

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 $R$ of elements of the free basis.

Every free module is also a projective module, as well as a flat module.