A module over a unit ring is called divisible if, for all which are not zero divisors, every element of can be "divided" by , in the sense that there is an element in such that . This condition can be reformulated by saying that the multiplication by defines a surjective map from to .
It can be shown that every injective -module is divisible, but the converse only holds for particular classes of rings, e.g., for principal ideal domains. Since and are evidently divisible -modules, this allows us to conclude that they are also injective.
An additive Abelian group is called divisible if it is so as a -module.