Ozone Groups and Centers of Skew Polynomial Rings

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 polynomial identity (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 ring to be Gorenstein (resp. regular) in low dimension.