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 
.
