Non-associative versions of Hilbert's basis theorem
Per Bäck, Johan Richter
TL;DR
This work extends Hilbert's basis theorem to non-associative algebra by establishing Noetherian properties for non-associative skew Laurent polynomial rings, non-associative Ore extensions, and their skew power series and Laurent series generalizations under precise conditions on the maps $\sigma$ and $\delta$. A key finding is that left and right Noetherianity diverge in the non-associative setting, with a robust right-Noetherian result for $R[X;\sigma,\delta]$ and a left-counterexample, alongside full left/right results for $R[X^{\pm};\sigma]$. The results for the power-series and Laurent-series analogues further broaden the landscape, showing right Noetherian behavior under suitable hypotheses and highlighting open questions about left versions. Collectively, the paper generalizes BR22, clarifies non-associative obstacles, and deepens understanding of Noetherian phenomena in non-associative Ore-like constructions, with potential implications for non-associative geometry and representation theory.
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.