Radicals and Köthe’s Conjecture for Skew PBW Extensions

Abstract

The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.