Hilbert's Basis Theorem for generalized nonassociative Ore extensions
Per Bäck, Masood Aryapoor
TL;DR
The paper broadens Hilbert’s Basis Theorem to a wide class of nonassociative, noncommutative polynomial rings by introducing generalized nonassociative Ore extensions (GNOEs) and their flipped variants. It formalizes left and right GNOEs, develops left and right Euclidean division algorithms, and proves that Noetherianity ascends from a base ring to its GNOE extension when division algorithms hold. By linking Euclidean division to Noetherianity, the authors establish a unified framework that yields Hilbert-type results for both standard and flipped nonassociative Ore constructions, including $R[X;\sigma,\delta]$ and Cayley-type doubles. The work also provides explicit sufficient conditions on $\sigma$ and $\delta$ under which Euclidean division algorithms exist, thereby enabling practical transfer of Noetherian properties to new nonassociative 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.