Firstly note that holds. Thus, I shall prove that , which will prove the result.

Let us forget about rings for a moment and look at . This is clearly equal to . So,

. As this is always a whole number and as then either is even with or and .

Thus, , . This expands to .

Therefore, . Inserting this into our ring we see that this is actually just equal to , as required.

Thus, by Problem 36, every element is central and so the ring is commutative.