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Possible Duplicate:
Ring such that $x^4=x$ for all $x$

Let $R$ be a ring such that $a^4=a$ $ ,\forall a \in R$. How do I show that $R$ is commutative?

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    I answered this here: http://math.stackexchange.com/questions/76792/ring-such-that-x4-x-for-all-x/77016#770162011-11-16

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If 2 is invertible, it may work this way:

For arbitrary elements $a,b$ calculate $ab-ba = (ab-ba)^4 = (ba-ab)^4 = ba-ab$. So $2ab=2ba$ and hence $ab=ba$.

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    no, it's a good try. I think it's not easy to prove it, maybe I am not good at algebra.2011-11-16