Radicals in Ore extensions

Abstract

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.