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 .