Suppose is a ring, is an -module, and is a submodule. Then there is an -module together with a module morphism , such that and any module morphism with factors uniquely through :
This might look crazy at first glance, but once we cover universal properties in category theory we will see how it captures the true essence of the quotient module construction. For now, we simply note the following comforting facts.
First, the abelian group structure on is the usual group quotient of the abelian group by the (automatically normal) subgroup . As such, its elements correspond to the distinct cosets of in . We can then always represent an element of in the form for some , with the understanding that two cosets and define the same element in exactly when . Moreover, the additive operation on is exactly coset addition: .
Second, the action of on the is inherited from the action of on . In other words, for and in we have . Convince yourself this is actually well defined!
The projection map is the same as the projection map for the abelian groups, i.e., . The key fact is that this is an -module morphism: .
The fact that is immediate from the same result for groups and the definition of the kernel for a module morphism.
The universal property is then almost immediate from the corresponding universal property for quotient groups in the category .