Ozone groups and centers of skew polynomial rings

TL;DR

This paper introduces the ozone group, an automorphism group fixing the center of a noncommutative algebra, and studies its properties in PI skew polynomial rings to determine conditions for the center to be Gorenstein or regular.

Contribution

It defines the ozone group and explores its role in understanding the structure of PI skew polynomial rings, providing explicit criteria for the center's Gorenstein and regular properties.

Findings

Ozone group is introduced as automorphisms fixing the center.

Explicit conditions for the center to be Gorenstein.

Explicit conditions for the center to be regular.

Abstract

We introduce the ozone group of a noncommutative algebra $A$, defined as the group of automorphisms of $A$ which fix every element of its center. In order to initiate the study of ozone groups, we study PI skew polynomial rings, which have long proved to be a fertile testing ground in noncommutative algebra. Using the ozone group and other invariants defined herein, we give explicit conditions for the center of a PI skew polynomial to be Gorenstein (resp. regular) in low dimension.