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 .

Griffith, P. A. Infinite Abelian Group Theory. Chicago, IL: University of Chicago Press, p. 21,
1970.Kargapolov, M. I. and Merzljakov, Yu. I. "Rank of an
Abelian Group." §7.2 in Fundamentals
of the Theory of Groups. New York: pp. 53-54, 1979.Munkres,
J. R. Elements
of Algebraic Topology. Menlo Park, CA: Addison-Wesley, p. 24, 1984.Schenkman,
E. "Rank and Linear Independence." §2.4 in Group
Theory. Princeton, NJ: Van Nostrand, pp. 51-56, 1965.