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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?

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    Can you find two matrices with integer entries for which $AB \neq BA$?2011-05-21
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    If you pick two endomorphisms of $\mathbb Z$ you will find they do commute (do do this). So let's do the “next” case: if you pick a pair of endomorphisms of $\mathbb Z\oplus\mathbb Z$ more or less however you want, you will find that they don't commute. It is just a matter of trying until you find an example. You *should* try!2011-05-21

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