Transparent Ore extensions over weak $\sigma$-rigid rings

Recall that a Noetherian ring $R$ is said to be a Transparent ring if there exist irreducible ideals $I_j$, $1\leq j\leq n$ such that $\bigcap_{j=1}^n I_j = 0$ and each $R/I_j$ has a right Artinian quotient ring. Let $R$ be a commutative Noetherian ring, which is also an algebra over $\mathbb Q$ (the field of rational numbers); $\sigma$ an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. Also let $R$ be a weak $\sigma$-rigid ring (i.e. $a\sigma(a)\in N(R)$ if and only if $a\in N(R)$, where $N(R)$ the set of nilpotent elements of R). Then we prove that $R[x;\sigma,\delta]$ is a Transparent ring.