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

Johan Richter

TL;DR

This work extends centralizer theory to non‑associative algebras by introducing a pseudo-degree framework that generalizes Amitsur‑type results to elements in the nucleus. The main result shows that for an element $a$ with positive pseudo-degree in $N_l(S)\cup N_m(S)\cup N_r(S)$ and with $C_S(a)$ satisfying a $D(\ell)$ condition, the centralizer $C_S(a)$ is a finite free left $K[a]$-module, with rank bounded by $\ell m$ (where $m=\nabla(a)$) and $K[a]\cong K[x]$. The approach yields algebraic dependence results for commuting elements and applies to non‑associative Ore extensions and octonionic constructions, illustrating the theory beyond associative settings. Overall, the paper provides a principled method to bound and compute centralizers in nuclei of non‑associative algebras, expanding centralizer theory to new algebraic contexts.

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 $a$.

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

TL;DR

This work extends centralizer theory to non‑associative algebras by introducing a pseudo-degree framework that generalizes Amitsur‑type results to elements in the nucleus. The main result shows that for an element with positive pseudo-degree in and with satisfying a condition, the centralizer is a finite free left -module, with rank bounded by (where ) and . The approach yields algebraic dependence results for commuting elements and applies to non‑associative Ore extensions and octonionic constructions, illustrating the theory beyond associative settings. Overall, the paper provides a principled method to bound and compute centralizers in nuclei of non‑associative algebras, expanding centralizer theory to new algebraic contexts.

Abstract

This papers studies centralizers of an element, , in the nucleus of a non-associative algebra with a special type of valuation. We prove that the centralizer of is a free module of finite rank over the algebra generated by .
Paper Structure (3 sections, 13 theorems, 17 equations)

This paper contains 3 sections, 13 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Centralizers in algebras with degree functions

Key Result

Theorem 1.1

Let $P=\sum_{i=0}^n p_i D^i$ and $Q= \sum_{j=0}^{m} q_j D^j$ be two commuting differential operators, with polynomial coefficients and constant leading coefficients. Then there is a non-zero polynomial $f(s,t)$ in two commuting variables over $\mathbb{C}$ such that $f(P,Q) =0$. Note that the fact th

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 18 more