[1] The rank of A determines the size of the largest free abelian group contained in A.
If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A.
The term rank has a different meaning in the context of elementary abelian groups.
A subset {aα} of an abelian group A is linearly independent (over Z) 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 (so that the sum is, in effect, finite), then all coefficients are zero.
Any two maximal linearly independent sets in A have the same cardinality, which is called the rank of A.
For instance, for every cardinal d there 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.
[citation needed] Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.
into k natural summands, the group A is the direct sum of k indecomposable subgroups of ranks
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 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 (or, to be more specific, vector space) over it is free.
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,