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?
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 $ab=ba$.