Why is the endomorphism ring of $\mathbb{Z}\times\mathbb{Z}$ noncommutative? 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?
 A: 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.
A: An endomorphism of $\mathbb{Z}\times\mathbb{Z}$ can be represented as a $2\times2$ matrix with entries in $\mathbb{Z}$, and all such matrices give endomorphisms of $\mathbb{Z}\times\mathbb{Z}$.  (The ones with determinant 1 give automorphisms.)
So any two $2\times 2$ matrices that don't commute will provide you with an example.
A: HINT $\rm\ \ \ a\ x + y\:\ \ne\:\ a\ (x+y)\ \ $ if $\rm\ \ a\ne 1$ 
$\quad\ $ i.e. $\rm\ \ \ \ \ (x,\:y)\ \mapsto\ (a\:x,\:y) $
$\quad$ and $\rm\ \ \ \ \:(x,\:y)\ \mapsto\ (x+y,\:y)\ \ $ do not commute.
A: Hint: try to show that the ring $M_2(\mathbb{Z})$ of $2 \times 2$ matrices with $\mathbb{Z}$-coefficients acts effectively faithfully on $\mathbb{Z}^2$ by endomorphisms.  This is a non-commutative ring, so this will answer your question.
More generally, if $A$ is an abelian group and $n \in \mathbb{Z}^+$, then one can show that $\operatorname{End}(A^n) = M_n(\operatorname{End}(A))$.  (It happens that the endomorphism ring of the group $\mathbb{Z}$ is canonically isomorphic to the ring $\mathbb{Z}$, so the previous paragraph is a special case of this.)  This gives many examples of non-commutative endomorphism rings.
A: Every ring acts by left multiplication as endomorphisms of its underlying abelian group ("Cayley's theorem for rings"), so if you believe that there exist non-commutative rings, then you already believe that there exist non-commutative endomorphism rings. 
