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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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