Prime radical of Ore extensions over $\delta$-rigid rings

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).