Centrifiers and ring commutativity

A result of Herstein says in particular that if there exists n > 1 such that xᵑ − x ∈ Z(R) for all x in a ring R then R is commutative. We give an elementary proof of this fact for certain values of n, based on the theory of centrifiers which we develop. For n = 5; 7, we also give an elementary proof of the commutativity of rings R such that xᵑ + x ∈ Z(R) for all x ∈ R