A subset of an abelian group is linearly independent if the only linear combination of these elements that is equal to zero is trivial: if where all but finitely many coefficients nα are zero, then all summands are 0. Any two maximal linearly independent sets in A have the same cardinality, which is called the rank of A. Rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group A is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted T. A group is called torsion-free if it has no non-trivial torsion elements. The factor-group A/T is the unique maximal torsion-free quotient of A and its rank coincides with the rank of A. The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponding to modules over Z. For this, see finitely generated module#Generic rank.
Properties
The rank of an abelian group A coincides with the dimension of the Q-vector space A ⊗ Q. If A is torsion-free then the canonical map A → A ⊗ Q is injective and the rank of A is the minimum dimension of Q-vector space containing A as an abelian subgroup. In particular, any intermediate groupZn < A < Qn has rank n.
The group Q of rational numbers has rank 1. Torsion-free abelian groups of rank 1 are realized as subgroups of Q and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2.
Rank is additive over short exact sequences: if
Rank is additive over arbitrary direct sums:
Groups of higher rank
Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal dthere exist torsion-free abelian groups of rank d that are indecomposable, i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well understood. Moreover, for every integer, there is a torsion-free abelian group of rank that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups. Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined. Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers, there exists a torsion-free abelian group A of rank n such that for any partition into k natural summands, the group A is the direct sum of k indecomposable subgroups of ranks. Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of A. Other surprising examples include torsion-free rank 2 groups An,m and Bn,m such that An is isomorphic to Bn if and only if n is divisible by m. For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that
K is indecomposable;
K is generated by G and a single other element; and
Every nonzero direct summand of G is decomposable.
Generalization
The notion of rank can be generalized for any module M over an integral domain R, as the dimension over R0, the quotient field, of the tensor product of the module with the field: It makes sense, since R0 is a field, and thus any module over it is free. It is a generalization, since any abelian group is a module over the integers. It easily follows that the dimension of the product over Q is the cardinality of maximal linearly independent subset, since for any torsion element x and any rational q
