In mathematics, a free module is a module that has a basis, that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.

Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S.

A free abelian group is precisely a free module over the ring ${\displaystyle \mathbb {Z} }$ of integers.

For a ring ${\displaystyle R}$ and an ${\displaystyle R}$-module ${\displaystyle M}$, the set ${\displaystyle E\subseteq M}$ is a basis for ${\displaystyle M}$ if:

A free module is a module with a basis.

An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of M.

If ${\displaystyle R}$ has invariant basis number, then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number. The cardinality of any (and therefore every) basis is called the rank of the free module ${\displaystyle M}$. If this cardinality is finite, the free module is said to be free of finite rank, or free of rank n if the rank is known to be n.

Let R be a ring.