TITLE: Why is the endomorphism ring of $\mathbb{Z}\times\mathbb{Z}$ noncommutative?
QUESTION [6 upvotes]: So I hear that the endomorphism ring of an abelian group is not always commutative. In particular, I'm looking at the abelian group $A=\mathbb{Z}\times\mathbb{Z}$, and considering $\text{End } A$. I can't find a counterexample to show that $\text{End } A$ is not commutative. Does anyone know of two such endomorphisms that don't commute?

REPLY [8 votes]: I know other people have given good answers but here's an explicit example:
$\phi : (a,b) \mapsto (b,a)$ and $\theta: (a,b) \mapsto (a+a,b)$
both are homomorphisms from $A$ to $A$, but the order you apply them matters.