Let $R$ be a ring , $n \gt 1$, such that for all $x \in R$: $x^n-x \in Z(R)$, the center of $R$. Does it follow that $R$ is commutative?


For $n=2,3$ this is pretty straightforward to prove. But what about higher powers? This is a generalization of Nathan Jacobson's theorem that if $x^n=x$ for all $x \in R$, then $R$ is commutative.



Ring in which $x^n-x$ is central for every $x$
Herstein, "A generalization of a theorem of Jacobson" Amer. J. Math. 73 (1951), 756–762 proves this. Also part III, Amer. J. Math. 75 (1953), 105–111 proves something a bit more general. Namely, the hypothesis can be weakened to $\forall x\ \exists\,n>1\ \cdots$


