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No nilpotent elements

Posted: Thu May 19, 2016 10:26 am
by Tolaso J Kos
Let $\mathcal{R}$ be a ring such that:

$$\text{there exists an $n \geq 2$ such that: $a^n =a$ forall $a \in \mathcal{R}$}$$

Prove that $\mathcal{R}$ has no zero nilpotent elements. Furthemore, if $n$ is even prove that the characteristic of $\mathcal{R}$ is $2$.