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PI degree parity in $$q$$-skew polynomial rings. (English) Zbl 1153.16025
Suppose that an associative algebra $$A$$ is an iterated skew polynomial extension $$k[x_1;\tau_1,\delta_1]\cdots[x_n;\tau_n,\delta_n]$$ such that $$\tau_i(x_j)=\lambda_{ij}x_j$$ for all $$j<i$$ and $$\delta_i\tau_i=q_i\tau_i\delta_i$$ where $$\lambda_{ij},q_i\in k^*$$ and $$q_i\neq 1$$. It is also assumed that each $$\tau_i$$-derivation $$\delta_i$$ extends to a locally nilpotent derivation on $$k\langle x_1,\dots,x_{i-1}\rangle$$. Under these assumptions it is shown that $$A$$ is a PI-algebra if and only if all $$\lambda_{ij}$$ are roots of 1. Clearly in this case the PI-degree of $$A$$ is the same as the PI-degree of the related quantum polynomial algebra with multiparameters $$\lambda_{ij}$$.

##### MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16R50 Other kinds of identities (generalized polynomial, rational, involution)
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