Ideal Krull-symmetry of iterated extensions

A ring R is said to be ideal Krull-symmetric if for any ideal I of R, the right Krull dimension of I is equal to the left Krull dimension of I. Let now R be commutative Noetherian ring. In this paper we show that certain Ore extensions of R are ideal Krull-symmetric. The rings that we deal with are:

• $S_t(R)=R[x_1;\sigma_1][x_2;\sigma_2]\dots[x_t,\sigma_t]$, the iterated skew-polynomial ring, where each $\sigma_i$ is an automorphism of $S_{i-1}(R)$
• $L_t(R) = R[x_1, x_1^{-1}; \sigma_1][x_2, x_2^{-1};\sigma_2\dots[x_t,x_t^{-1};\sigma_t]$, the iterated skew-Laurent polynomial ring, where each $\sigma_i$ is an automorphism of $L_{i-1}(R)$
• $D_t(R) = R[x_1;\delta_1][x_2;\delta_{2}]\dots[x_t;\delta_t]$, the iterated differential polynomial ring, where each $\delta_i$ is a derivation of $D_{i-1}(R)$ such that each $\delta_i\mid R$ is a derivation of R and,
• $A_t(R)$ is any of $S_t(R)$ or $L_t(R)$, where $\sigma_i\mid R$ is an automorphism of R.

With this we prove that $A_t(R)$ and $D_t(R)$ are ideal Krull-symmetric.