Radicals of skew polynomial rings and skew Laurent polynomial rings

We first introduce the σ-Wedderburn radical and the σ-Levitzki radical of a ring R, where σ is an automorphism of R. Using the properties of these radicals, we study the Wedderburn radical of the skew polynomial ring R[x;σ] and the skew Laurent polynomial ring R[x,x−1;σ], and next observe the Levitzki radical of R[x;σ] and R[x,x−1;σ]. Furthermore we characterize the upper nilradical of R[x;σ] and R[x,x−1;σ], via the upper σ-nil radical of R.