Radicals of Ore Extension of Skew Armendariz Rings

Abstract

Let R be a ring with an endomorphism α and an α-derivation δ. In this article, for a skew-Armendariz ring R we study some properties of skew polynomial ring R[x; α, δ]. In particular, among other results, we show that for an (α, δ)-compatible skew-Armendariz ring R, γ(R[x; α, δ]) = γ(R)[x; α, δ] = Niℓ*(R)[x; α, δ], where γ is a radical in the class of radicals which includes the Wedderburn, lower nil, Levitzky, and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip, and 2-primal property, transfer between R and the skew polynomial ring R[x; α, δ], in case R is (α, δ)-compatible skew-Armendariz. As a consequence we extend and unify several known results.