How a nonassociative algebra reflects the properties of a skew polynomial

TL;DR

This paper explores how nonassociative algebras constructed from skew polynomials reflect their properties, providing criteria for irreducibility and analyzing their structure, especially the right nucleus, in relation to polynomial factors.

Contribution

It introduces a new construction linking skew polynomials to nonassociative algebras and characterizes their structure and irreducibility criteria.

Findings

Construction of nonassociative algebras from skew polynomials

Criteria for irreducibility of low-degree skew polynomials

Analysis of the right nucleus structure in relation to polynomial factors

Abstract

Let $S$ be a unital associative ring and $S[t;\sigma,\delta]$ be a skew polynomial ring, where $\sigma$ is an injective endomorphism of $S$ and $\delta$ a left $\sigma$-derivation. For each $f\in S[t;\sigma,\delta]$ of degree $m>1$ with a unit as leading coefficient, we construct a unital nonassociative algebra whose behaviour reflects the properties of $f$. The algebras obtained yield canonical examples of right division algebras when $f$ is irreducible. We investigate the structure of these algebras. The structure of their right nucleus depends on the choice of $f$. In the classical literature, this nucleus appears as the eigenspace of $f$, and is used to investigate the irreducible factors of $f$. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible.