Let $A$ be a free associative unital algebra with some set of generators over a field $k$. Then the centralizer or any nonscalar element $u\in A$ is isomorphic to $k[z]$ for some nonscalar element $z\in A$

As a corollary, any two commuting elements in $A$ may be presented as values of two polynomials $P(z), Q(z)$ over $k$ on the same element $z$ in $A$.

Literature

This has been conjectured by P. M. Cohn and proved in

George M. Bergman, Centralizers in free associative algebras, Trans. Amer. Math. Soc. 137:327–344 (1969) doi

A proof using quantization is in

Alexei Kanev Belov?, Farrokh Razavinia, Wenchao Zhang, Bergman’s centralizer theorem and quantization, Communications in Algebra 46:5 (2018) doiarXiv:1708.04802

A combinatorial approach is in chapter 3 of

Zlil Sela, Noncommutative algebraic geometry, I: Monomial equations with a single variable, Model theory 3:3 (2024) 733–800 doi