On 2-primal Ore extensions over Noetherian $\sigma(*)$-rings

In this article, we discuss the prime radical of skew polynomial rings over Noetherian rings. We recall $\sigma(*)$ property on a ring $R$ (i.e. $a\sigma(a)\in P(R)$ implies $a\in P(R)$ for $a\in R$, where $P(R)$ is the prime radical of $R$, and $\sigma$ an automorphism of $R$). Let now $\delta$ be a $\sigma$-derivation of $R$ such that $\delta(\sigma(a))=\sigma(\delta(a))$ for all $a\in R$. Then we show that for a Noetherian $\sigma(*)$-ring, which is also an algebra over $\mathbb Q$, the Ore extension $R[x;\sigma,\delta]$ is 2-primal Noetherian (i.e. the nil radical and the prime radical of $R[x;\sigma,\delta]$ coincide).