Again, once we have a new structure it’s important to understand what sort of functions connect different instancees of the structure. For modules we have module homomorphisms.
By now this should be pretty straightforward. A left -module is an abelian group equipped with an action of . A homomorphism of -modules is a homomorphism of abelian groups that also preserves the action of . That is, it’s a linear function satisfying for all . We call such a function -linear.
In terms of the tensor product picture, any linear function gives us a linear function . The condition of -linearity is that the following diagram commute:
where the horizontal arrows are the actions of on and , respectively.
Now if we pick two left -modules and we have a bunch of different -module homomorphisms from to . We’ll call the set of them , sometimes shortened to , or even just . The important thing about this set is that it inherits the structure of an abelian group from the one on .
There’s always a homomorphism sending every element of to the zero element of . Also, given homomorphisms and we can define . It’s straightforward to check that this preserves the action of as long as and both do. Finally, there’s a homomorphism defined by . We can easily see that these operations on satisfy the axioms of an abelian group.
In fact it gets even better. Not only are the homomorphism sets abelian groups, but composition is bilinear! Let’s consider four homomorphisms between three modules , , , and . Then we build up the composition . Let’s see what it does to an element .
This gives us a linear function for every three modules , , and .
What happens if we pick all three modules to be the same one? Each homomorphism set is , which we’ll call . Then we get a linear function . This is a ring structure! We call it the ring of -endomorphisms of . If the ring is the ring of integers and is an abelian group, then is just the endomorphism ring we considered earlier. This is an example of how the theory of modules naturally extends the theory of abelian groups.