Radicals in polynomial rings skewed by an endomorphism

Abstract

Given a ring R, radicals of the polynomial ring $R\left[x\right]$, and even of the skew polynomial ring $R[x;\sigma ]$ skewed by an endomorphism σ on R, have been studied and described for many different radicals. Often, in those descriptions, the ring R was assumed to be unital, or the endomorphism σ was assumed to be an automorphism. Here we systematically study what happens when such assumptions are dropped, and generalize to even more radicals. Our results reveal three key properties that, when present, allow a simple description of a radical of $R[x;\sigma ]$. Moreover, multiple examples are provided showing that when σ is injective, but not surjective, wild growth patterns may occur.

We also answer an open question in the literature, by showing that the Levitzki radical of the skew Laurent polynomial ring $R[x,x^{-1};\sigma ]$ (when σ is an automorphism) is not naively describable in terms of the Levitzki radicals of $R[x;\sigma ]$ and $R[x;{\sigma }^{-1}]$.