Rank of an abelian group

Summary

The rank of an abelian group is the largest number of its elements that are independent over the integers. It equals the dimension of the rational vector space obtained by tensoring the group with $\Q$. It plays for abelian groups the role that dimension plays for vector spaces, but it ignores torsion: finite groups have rank $0$. For a finitely generated abelian group, the rank is the number of copies of $\Z$ in its decomposition.