Ideal Krull-symmetry of skew polynomial rings

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Let R be a commutative Noetherian ring and σ an automorphism of R. Let K be a commutative Noetherian ring, which is also an algebra over \({\mathbb{Q}}\) , and δ a derivation of K (\({\mathbb{Q}}\) is the field of rational numbers). Consider the rings R[x; σ], R[x, x −1; σ] and K[x; δ]. We show that if I is a two-sided ideal in any one of these rings, then the left and right Krull dimensions of I coincide.