Prove that for any element a of a finite ring R,some power a^n (with n>=1)is an idempotent? finite?

For any $r\in R$,
since $R$ is finite,
the elements in the set $\{ r, r^2, r^3, ...\}$
can't be all distinct.
Suppose $r^a=r^{a+b}$ for some positive integer $a$ and $b$.
Then
$(x^{ab})^2
=(x^{ab})(x^{ab})
=x^{ab-a}\cdot x^a \cdot x^{\overbrace{b+b+\cdots +b}^{a \mbox{ times}}}
=x^{ab-a}\cdot (x^a \cdot x^b) \cdot x^{\overbrace{b+b+\cdots +b}^{a-1 \mbox{ times}}}
=x^{ab-a}\cdot x^{a+b} \cdot x^{\overbrace{b+b+\cdots +b}^{a-1 \mbox{ times}}}
=x^{ab-a}\cdot x^{a} \cdot x^{\overbrace{b+b+\cdots +b}^{a-1 \mbox{ times}}}
=\cdots =x^{ab-a}\cdot x^a=x^{ab}$.

$x^{ab}$ is an idempotent element.