For any prime number 
and any positive integer 
,
the 
-rank 
of a finitely generated Abelian
group 
is the number of copies of the cyclic group 
appearing in the Kronecker decomposition
of 
(Schenkman 1965). The free (or torsion-free)
rank 
of 
is the number of copies of 
appearing in the same decomposition. It can be characterized
as the maximal number of elements of 
which are linearly independent over 
. Since it is also equal to the dimension of 
as a vector space
over 
, it is often called the rational
rank of 
.
Munkres (1984) calls it the Betti number of 
.
Most authors refer to 
simply as the "rank" of 
(Kargapolov and Merzljakov 1979), whereas others (Griffith
1970) use the word "rank" to denote the sum 
. In this latter meaning, the rank
of 
is the number of direct summands appearing
in the Kronecker decomposition of 
.
