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In mathematics, the **rank**, **Prüfer rank**, or **torsion-free rank** of an abelian group *A* is the cardinality of a maximal linearly independent subset.^{[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*. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.

The term rank has a different meaning in the context of elementary abelian groups.

A subset {*a*_{α}} 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 *n*_{α} 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*.

The 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*). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group *A*/*T*(*A*) 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.

- 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 group**Z**^{n}<*A*<**Q**^{n}has rank*n*. - Abelian groups of rank 0 are exactly the periodic abelian groups.
- 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.^{[2]} - Rank is additive over short exact sequences: if

- is a short exact sequence of abelian groups then rk
*B*= rk*A*+ rk*C*. This follows from the flatness of**Q**and the corresponding fact for vector spaces.

- Rank is additive over arbitrary direct sums:

- where the sum in the right hand side uses cardinal arithmetic.

Abelian groups of rank greater than 1 are sources of interesting examples. 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. 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.^{[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.

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 .^{[citation needed]} 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 *A*_{n,m} and *B*_{n,m} such that *A*^{n} is isomorphic to *B*^{n} 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.

The notion of rank can be generalized for any module *M* over an integral domain *R*, as the dimension over *R*_{0}, the quotient field, of the tensor product of the module with the field:

It makes sense, since *R*_{0} is a field, and thus any module (or, to be more specific, vector space) over it is free.

It is a generalization, since every 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*,

**^**Page 46 of Lang, Serge (1993),*Algebra*(Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001**^**Thomas, Simon; Schneider, Scott (2012), "Countable Borel equivalence relations", in Cummings, James; Schimmerling, Ernest (eds.),*Appalachian Set Theory: 2006-2012*, London Mathematical Society Lecture Note Series, vol. 406, Cambridge University Press, pp. 25–62, CiteSeerX 10.1.1.648.3113, doi:10.1017/CBO9781139208574.003, ISBN 9781107608504. On p. 46, Thomas and Schneider refer to "...this failure to classify even the rank 2 groups in a satisfactory way..."