Noether's normalization in iterated skew polynomial rings

Abstract

The classical Noether Normalization Lemma states that if $S$ is a finitely generated algebra over a field $k$, then there exist elements $x_1,\dots ,x_n$ 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 flavors. 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 $D[x_1,\dots ,x_n;{\sigma }_1,\dots ,{\sigma }_n]$. In this paper, we investigate this lemma in a more general setting when $S$ is a quotient ring of an iterated skew polynomial ring $D[x_1;{\sigma }_1,{\delta }_1]\dots [x_n;{\sigma }_n,{\delta }_n]$. We extend several key results of Elad Paran and Thieu N. Vo to this broader context and introduce a new version of Combinatorial Nullstellensatz over division rings.