Editing Divisible group (section)

==Reduced abelian groups==

An abelian group is said to be '''reduced''' if its only divisible subgroup is {0}. Every abelian group is the direct sum of a divisible subgroup and a reduced subgroup.  In fact, there is a unique largest divisible subgroup of any group, and this divisible subgroup is a direct summand.<ref>Griffith, p.7</ref> This is a special feature of [[hereditary ring]]s like the integers '''Z''': the [[direct sum of modules|direct sum]] of injective modules is injective because the ring is [[Noetherian ring|Noetherian]], and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective.  The converse is a result of {{harv|Matlis|1958}}: if every module has a unique maximal injective submodule, then the ring is hereditary.

A complete classification of countable reduced periodic abelian groups is given by [[Ulm's theorem]].