free module

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

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

This is equivalent to sayingthat the module has a free basis,i.e. a set of elementswith the propertythat every element of the modulecan be uniquely expressedas 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.