On σ-nil ideals of bounded index of σ-nilpotence

We investigate properties of σ-nil subsets with bounded index of σ-nilpotence, beginning with a classification of many of the possible types of nilpotence available in the σ-skewed case. In the process we introduce a σ-analog of the bounded nilradical of a ring. In many situations we completely describe the bounded nilradical for skew polynomial rings in terms of the σ-bounded nilradical in the coefficient ring. We also construct an example demonstrating that the bounded nilradical for skew polynomial rings is more complicated than one might initially guess. Further extensions are explored when one assumes additional restrictions on σ.