Today I want to set out an incredibly important example of a ring. This example (and variations) come up over and over and over again throughout mathematics.
Let’s start with an abelian group . Now consider all the linear functions from back to itself. Remember that “linear function” is just another term for “abelian group homomorphism” — it’s a function that preserves the addition — and that we call such homomorphisms from a group to itself “endomorphisms”.
As for any group, this set has the structure of a monoid. We can compose linear functions by, well, composing them. First do one, then do the other. We define the operation by and verify that the composition is again a linear function:
This composition is associative, and the function that sends every element of to itself is an identity, so we do have a monoid.
Less obvious, though, is the fact that we can add such functions. Just add the values! Define . We check that this is another endomorphism:
Now this addition is associative. Further the function sending every element of to the element of is an additive identity, and the function is an additive inverse. The collection of endomorphisms with this addition becomes an abelian group.
So we have two structures: an abelian group and a monoid. Do they play well together? Indeed!
showing that composition distributes over addition.
So the endomorphisms of an abelian group form a ring with unit. We call this ring , and like I said it will come up everywhere, so it’s worth internalizing.