PI degree parity in q-skew polynomial rings

For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x1;τ1,δ1] [xn;τn,δn] agrees with the PI degree of R[x1;τ1] [xn;τn] when each (τi,δi) satisfies a qi-skew relation for qi k× and extends to a higher qi-skew τi-derivation. We confirm the quantum Gelʹfand–Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras.