A note on a theorem of Jacobson related to periodic rings

We show that if $R$ is a ring such that for each $x\in R$ there exist two natural numbers $n(x)$ and $m(x)$ of opposite parity with $x^{n(x)}=x^{m(x)}$, then $R$ is commutative. This extends the classical famous theorem of Jacobson [Ann. of Math. 46 (1945), p. 695-707] for commutativity of potent rings.