Centralizers in non-associative rings with a pseudo-degree function

TL;DR

This paper investigates the structure of centralizers in non-associative algebras equipped with a pseudo-degree function, revealing they form finite-rank free modules over the algebra generated by a given element.

Contribution

It introduces a novel approach using pseudo-degree functions to analyze centralizers, establishing their finite free module structure in non-associative algebras.

Findings

Centralizers are finite-rank free modules over the algebra generated by an element.

The study extends understanding of algebraic structure in non-associative settings.

Provides new tools for analyzing valuations in non-associative algebras.

Abstract

This papers studies centralizers of an element, $a$, in the nucleus of a non-associative algebra with a special type of valuation. We prove that the centralizer of $a$ is a free module of finite rank over the algebra generated by $a$.