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Let $A$ be a ring. Let $E$ be the set of polynomials $\{X^n-X \in \mathbb{Z}[X]|n \in \mathbb{N}^*-\{1\}\}$.

By the theorem of Jacobson, we know that if for each $a\in A$ there is an element of $E$ for which $a$ is a root, then $A$ is commutative.

Is there a characterization of the sets $F \subset \mathbb{Z}[X]$ such that, for all ring $A$, if every element of $A$ is a root of a polynomial in $F$, then $A$ is commutative ?

Thanks in advance.

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    @$f$rancis-jamet: oh, right. I'm thinking of an earlier version of the result, not the more general one. I fixed the statement. By the way, I just found [this MO thread](http://mathoverflow.net/questions/32032/on-a-theorem-of-jacobson) which is related.2012-06-27

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