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 pairs 
of integers, is a free Abelian group of rank 2.
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
.
