Radicals in Ore extensions

Let Formula: see text be a ring, Formula: see text be an automorphism of Formula: see text and Formula: see text be a Formula: see text-derivation of Formula: see text. We use Formula: see text to denote the set of all words composed of Formula: see text, Formula: see text and Formula: see text. A Formula: see text-ideal Formula: see text of Formula: see text is Formula: see text-prime if whenever Formula: see text are such that Formula: see text for any Formula: see text, we have Formula: see text or Formula: see text. In this paper, we first introduce the Formula: see text-prime ideal and the Formula: see text-prime radical of a ring Formula: see text, to obtain connections between the prime radical of the Ore extension Formula: see text and the Formula: see text-prime radical of the base ring Formula: see text. Based on these results, we next give definitions of the Formula: see text-LS-prime ideal, the Formula: see text-strongly prime ideal and the Formula: see text-uniformly strongly prime ideal of a ring Formula: see text to provide formulas for the LS-prime radical, the strongly prime radical and the uniformly strongly prime radical of the Ore extension.