A free Abelian group is an Abelian group 
with a basis 
such that every element
of 
has a unique expression as a finite sum of integer multiples of elements
of 
.
Equivalently,
 |
Thus a free Abelian group is a direct sum, rather than an infinite direct product,
of copies of the integers 
, and its group rank is the cardinal number of 
. For example, 
, whose elements are
ordered pairs 
of integers, is a free Abelian
group of group rank 2.
A free Abelian group is an Abelian group, but is not a free group (except in group
ranks zero and one, giving the trivial
group and 
,
respectively). Free Abelian groups are the free modules
in the case when the ring is the ring
of integers 
.
