Let $R$ be a ring such that every element is potent ($x^k = x$, for some integer $k>1$) or central. Prove that $R$ is commutative.
My prove:
Let $x,y$ be elements of $R$, suppose one of them is central (x is central or y is central) then it is obviously that $xy=yx$ because of property of central.
But how to prove that is $x^m = x$, and $y^n = y$ for some integer $m>1$ and $n>1$ then $xy=yx$?