left by a unimodular matrix (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of is equivalent with multiplying on the right by a unimodular matrix. The Smith normal form of is a matrix where and are unimodular, and is a matrix such that all non-diagonal entries are zero, the non-zero diagonal entries are the first ones, and is a divisor of for . The existence and the shape of the Smith normal proves that the finitely generated abelian group is the direct sum where is the number of zero rows at the bottom of (and also the rank of the group). This is the fundamental theorem of finitely generated abelian groups. The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums. Infinite abelian groups The simplest infinite abelian group is the infinite cyclic group . Any finitely generated abelian group is isomorphic to the direct sum of copies of and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of prime power orders. Even though the decomposition is not unique, the number , called the rank of , and the prime powers giving the orders of finite cyclic summands are uniquely determined. By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups in which the equation admits a solution for any natural number and element of , constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to and Prüfer groups for various prime numbers , and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group is a subgroup of an abelian group then admits a direct complement: a subgroup of such that . Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called reduced. Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsion-free groups, exemplified by