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Prime ideals in skew polynomial rings and quantized Weyl algebras. (English) Zbl 0779.16010
The author investigates the structure of skew polynomial rings of the form $$T = R[\theta;\sigma,\delta]$$, where $$\sigma$$ and $$\delta$$ are both nontrivial. The main focus is the analysis of the prime spectrum of the ring $$T$$. In the case that $$R$$ is commutative, the prime ideals of $$T$$ are classified, the strong second layer condition is established and it is shown that the rank of $$T/P$$ is bounded for $$P$$ in a given clique of prime ideals. The author then proceeds to an investigation of the special case of $$q$$-skew derivations, where $$\sigma\delta = q\delta\sigma$$, for some constant $$q$$. Many of the rings arising as quantum analogues of classical algebras can be represented as iterated $$q$$-skew polynomial extensions; so this is a case of wide interest. As an indication of the utility of the methods developed, the prime ideals and prime factors of the quantum Weyl algebras over a field are analyzed. Readers of this paper may also be interested in a forthcoming Memoir of the Am. Math. Soc., by the present author and E. S. Letzter, entitled “Prime ideals in skew and $$q$$-skew polynomial rings”.

##### MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16D25 Ideals in associative algebras 16P50 Localization and associative Noetherian rings
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