The Unapologetic Mathematician

Mathematics for the interested outsider

Endomorphism rings

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 G. Now consider all the linear functions from G 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 \left[f\circ g\right](x)=f(g(x)) and verify that the composition is again a linear function:
\left[f\circ g\right](x+y)=f(g(x+y))=f(g(x)+g(y))=
f(g(x))+f(g(y))=[f\circ g](x)+[f\circ g](y)
This composition is associative, and the function that sends every element of G 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 \left[f+g\right](x)=f(x)+g(x). We check that this is another endomorphism:
\left[f+g\right](x+y)=f(x+y)+g(x+y)=f(x)+f(y)+g(x)+g(y)=
f(x)+g(x)+f(y)+g(y)=[f+g](x)+[f+g](y)
Now this addition is associative. Further the function {}0 sending every element of G to the element {}0 of G is an additive identity, and the function \left[-f\right](x)=-f(x) 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!
\left[(f_1+g_1)\circ(f_2+g_2)\right](x)=\left[f_1+g_1\right](\left[f_2+g_2\right]((x))=
f_1(f_2(x)+g_2(x))+g_1(f_2(x)+g_2(x))=
f_1(f_2(x))+f_1(g_2(x))+g_1(f_2(x))+g_1(g_2(x))=
\left[f_1\circ f_2\right](x)+\left[f_1\circ g_2\right](x)+\left[g_1\circ f_2\right](x)+\left[g_1\circ g_2\right](x)=
\left[f_1\circ f_2+f_1\circ g_2+g_1\circ f_2+g_1\circ g_2\right](x)
showing that composition distributes over addition.

So the endomorphisms of an abelian group G form a ring with unit. We call this ring {\rm End}(G), and like I said it will come up everywhere, so it’s worth internalizing.

April 10, 2007 - Posted by John Armstrong | Ring theory | | 1 Comment

1 Comment »

  1. [...] 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 [...]

    Pingback by Homomorphisms of modules « The Unapologetic Mathematician | April 23, 2007

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