Hilbert's Basis Theorem for generalized nonassociative Ore extensions

TL;DR

This paper extends Hilbert's Basis Theorem to a new class of nonassociative Ore extensions, linking Euclidean division algorithms with Noetherian properties in nonassociative, noncommutative polynomial rings.

Contribution

It introduces a generalized class of nonassociative Ore extensions and proves Hilbert's Basis Theorem for them, connecting Euclidean division algorithms to Noetherianity.

Findings

Generalized Hilbert's Basis Theorem for nonassociative Ore extensions

Existence of Euclidean division algorithms implies Noetherianity

Extension to nonassociative, noncommutative polynomial rings

Abstract

We introduce a broader class of nonassociative Ore extensions that unifies and generalizes several earlier constructions. We prove generalizations of Hilbert's Basis Theorem for this class, showing that they arise immediately from the existence of Euclidean division algorithms. These results extend Hilbert's Basis Theorem to new families of nonassociative, noncommutative polynomial rings and establish a novel and direct connection between Euclidean division algorithms and the left and right Noetherianity of such rings.