By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups formula_1 in which the equation formula_163 admits a solution formula_164 for any natural number formula_44 and element formula_3 of formula_1, 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 formula_168 and Prüfer groups formula_169 for various prime numbers formula_73, and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group formula_1 is a subgroup of an abelian group formula_49 then formula_1 admits a direct complement: a subgroup formula_174 of formula_49 such that formula_176. 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.