Suppose that $R$ is a ring such that for any $x\in R$ there exists $1<n(x)\in \mathbb{N}$ such that $x^{n(x)}-x\in Z(R)$. Prove that $R$ is commutative or if it is not commutative, then the ideal generated by all additive commutators is null.

I wanted to prove it like Jacobson-Herstein theorem. So if the assertion is true for division rings, from density and subdirect product representation theorems, I can prove it for left primitive rings and next for semiprimitive rings. But I can't prove it for division rings and arbitrary rings from semiprimitive rings.