Non-associative versions of Hilbert's basis theorem

TL;DR

This paper extends Hilbert's basis theorem to various non-associative algebraic structures, demonstrating new versions and highlighting differences from the associative case.

Contribution

It introduces multiple non-associative versions of Hilbert's basis theorem for Ore extensions and skew polynomial rings, revealing asymmetries not seen in associative algebra.

Findings

Both left and right Hilbert's basis theorems hold for non-associative skew Laurent polynomial rings.

A right version of the theorem holds for non-associative Ore extensions.

Counterexample shows the left version does not hold for non-associative Ore extensions.

Abstract

We prove several new versions of Hilbert's basis theorem for non-associative Ore extensions, non-associative skew Laurent polynomial rings, non-associative skew power series rings, and non-associative skew Laurent series rings. For non-associative skew Laurent polynomial rings, we show that both a left and a right version of Hilbert's basis theorem hold. For non-associative Ore extensions, we show that a right version holds, but give a counterexample to a left version; a difference that does not appear in the associative setting.