A free Abelian group is a group G with a subset which generates the group G with the only relation being a b = b a. That is, it has no group torsion. All such groups are a direct product of the integers Z, and have rank given by the number of copies of Z. For example, Z*Z = {(n, m)} is a free Abelian group of rank 2. A minimal subset b_1, ..., b_n that generates a free Abelian group is called a basis, and gives G as G = Z b_1 + ... + Z b_n. A free Abelian group is an Abelian group, but is not a free group (except when it has rank one, i.e., Z). Free Abelian groups are the free modules in the case when the ring is the ring of integers Z.
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