Noether's normalization in generalized skew polynomial rings

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 $D[x_1,\dots,x_n;\sigma_1,\dots,\sigma_n]$. In this paper, we study this lemma for a more general case when $S$ is a quotient ring of a generalized skew polynomial ring $D[x_1;\sigma_1,\delta_1]\dots[x_n;\sigma_n,\delta_n]$. 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.