In fact is the filtered colimit of a diagram of cyclic groups and inclusions between them. In more detail, whenever , multiplication by induces an inclusion
For each prime the -Prüfer group , similarly formed as the colimit of groups and inclusions between them, embeds in . Furthermore, it follows from the Chinese remainder theorem? that the induced map
It is also a cogenerator in the category of abelian groups.
Proof: let be an abelian group. It suffices to check that for every there exists such that . But if is the cyclic subgroup generated by , then it is easy to find a map such that , and then we can extend to a map using injectivity of .
This means every abelian group embeds into an injective abelian group,
where is the group of -adic integers (this is connected with the direct sum decomposition of into Prüfer -groups).
For any ring , its algebraic dual , equipped with the left -module structure defined by , is an injective cogenerator in the category of left -modules. This follows easily from the corresponding property for .