Commutativity of centralizers in a coproduct of a free algebra and a polynomial algebra

TL;DR

This paper proves that the centralizer of a nonscalar element in the coproduct of a free algebra and a polynomial algebra is commutative, extending Bergman's centralizer theorem with a combinatorial proof.

Contribution

It establishes the commutativity of centralizers in a coproduct of free and polynomial algebras, providing a new combinatorial proof of a key algebraic property.

Findings

Centralizer of a nonscalar element in the coproduct is commutative.

The proof uses a reduction from Bergman's proof and combinatorial methods.

Extends Bergman's centralizer theorem to a broader algebraic setting.

Abstract

We show that the centralizer of a nonscalar element in the coproduct $k\langle X\rangle *k[Y]$ of a free associative algebra and a polynomial algebra over a given field is commutative. For $k\langle X \rangle$ this is part of Bergman's centralizer theorem. Our proof relies on a reduction given in Bergman's proof and is of combinatorial nature, employing a strict order structure of the coproduct monoid.