Noether's normalization in generalized skew polynomial rings

TL;DR

This work extends Noether normalization to quotients of generalized skew polynomial rings over division rings by introducing the notion of automorphic normalizability. It defines automorphically normalizable extensions, proves that under commutativity and infinitude conditions on the fixed center set $F$, the tuple $(\text{ω},\text{δ}_1;\dots;\text{ω},\text{δ}_n)$ is automorphically normalizable, and employs a new division-ring Combinatorial Nullstellensatz to realize non-vanishing polynomials that yield a monic polynomial in the last automorphic variable. Collectively, these results broaden Noether normalization to noncommutative settings, enabling finite left-module normal forms for quotients of generalized skew polynomial rings and offering new tools for structure theory in division-ring contexts.

Abstract

The classical Noether Normalization Lemma states that if $S$ is a finitely generated algebra over a field $k$, then there are $x_1,\dots,x_n$ which are algebraically independent over $k$ such that $S$ is a finite module over $k[x_1,\dots,x_n]$. This lemma has been studied intensively in different flavours. In 2024, Elad Paran and Thieu N. Vo successfully generalized this lemma for the case when $S$ is a quotient ring of the skew polynomial ring $D[x_1,\dots,x_n;σ_1,\dots,σ_n]$. In this paper, we study this lemma for a more general case when $S$ is a quotient ring of a generalized skew polynomial ring $D[x_1;σ_1,δ_1]\dots[x_n;σ_n,δ_n]$. We generalize a number of results obtained by Elad Paran and Thieu N. Vo to the new context and introduce a new version of Combinatorial Nullsellensantz over division rings.

Theorems & Definitions (48)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4: Combinatorial Nullstellensatz for division rings II
• Theorem 1.5: Combinatorial Nullstellensatz over division rings III
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.5