Prime radical of Ore extensions over \(\delta\)-rigid rings

Abstract

Let R be a ring. Let \(\sigma\) be an automorphism of R and \(\delta\) be a  \(\sigma\)-derivation of R. We say that R is a \(\delta\)-rigid ring if \(a\delta(a)\in P(R)\) implies \(a\in P(R)\), \(a \in R\); where P(R) is the prime radical of R. In this article, we find a relation between the prime radical of a \(\delta\)-rigid ring R and that of \(R[x,\sigma,\delta]\). We generalize the result for a Noetherian Q-algebra (Q is the field of rational numbers).