Noether's normalization in skew polynomial rings

TL;DR

This paper extends Noether's normalization lemma to polynomial rings over division algebras, identifies conditions under which it holds, and demonstrates cases where it fails in skew polynomial rings.

Contribution

It proves the normalization lemma for polynomial rings over division algebras and establishes conditions for its validity in skew polynomial rings.

Findings

Normalization holds for polynomial rings over division algebras.

Normalization may fail in skew polynomial rings with certain automorphisms.

A necessary and sufficient condition is provided when the division algebra is a field.

Abstract

We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if $I$ is a proper ideal of the ring $R=F[t_1,\ldots,t_n]$ of polynomials over a field $F$, then the quotient ring $R/I$ is a finite extension of a polynomial ring over $F$. We prove that the lemma holds when $R=D[t_1,\ldots,t_n]$ is the ring of polynomials in $n$ central variables over a division algebra $D$. We provide examples demonstrating that Noether's normalization may fail for the skew polynomial ring $D[t_1,\ldots,t_n;\sigma_1,\ldots,\sigma_n]$ with respect to commuting automorphisms $\sigma_1,\ldots,\sigma_n$ of $D$. We give a sufficient condition for $\sigma_1,\ldots,\sigma_n$ under which the normalization lemma holds for such ring. In the case where $D=F$ is a field, this sufficient condition is proved to be necessary.