On $2$-primal Ore extensions over Noetherian weak $\sigma$-rigid rings

Let $R$ be a ring, $\sigma$ an endomorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. In this article, we discuss skew polynomial rings over $2$-primal weak $\sigma$-rigid rings. We show that if $R$ is a $2$-primal Noetherian weak $\sigma$-rigid ring, then $R[x;\sigma,\delta]$ is a $2$-primal Noetherian weak $\overline\sigma$-rigid ring.