free module

Let $R$ be a commutative ring with unity. A free module over $R$is a (unital) module isomorphic to a direct sum of copies of $R$. In particular, as every abelian group is a $\mathbb{Z}$-module, a free abelian group is a direct sum of copies of $\mathbb{Z}$. This is equivalent to saying that the module has a free basis,i.e. a set of elements with the that every element of the module can be uniquely expressed as an linear combination over $R$of elements of the free basis. In the case that a free module over $R$ is a sum of finitely many copies of $R$, 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 $X$, the free $R$-module $F(X)$ on the set $X$ is equipped with a function $i:X\to F(X)$ satisfying the property that for any other $R$-module $A$ and any function $f:X\to A$, there exists a unique $R$-module map $h:F(X)\to A$ such that $(h\circ i)=f$.