A free Abelian group is a group with a subset which generates the group with the only relation being . That is, it has no group torsion. All such groups are a direct product of the integers , and have rank given by the number of copies of . For example, is a free Abelian group of rank 2. A minimal subset , ..., that generates a free Abelian group is called a basis, and gives as

A free Abelian group is an Abelian group, but is not a free group (except when it has rank one, i.e., ). Free Abelian groups are the free modules in the case when the ring is the ring of integers .