Prime ideals in skew polynomial rings and quantized Weyl algebras
type: Item
instance of: scholarly article
title: Prime ideals in skew polynomial rings and quantized Weyl algebras
zbMATH Open document ID: 0779.16010
DOI: 10.1016/S0021-8693(05)80036-5
author: Kenneth R. Goodearl
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: 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 \textit{E. S. Letzter}, entitled ``Prime ideals in skew and \(q\)-skew polynomial rings''.
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P50
zbMATH DE Number: 130638
zbMATH Keywords: skew polynomial rings
zbMATH Keywords: prime spectrum
zbMATH Keywords: prime ideals
zbMATH Keywords: strong second layer condition
zbMATH Keywords: clique of prime ideals
zbMATH Keywords: \(q\)-skew derivations
zbMATH Keywords: \(q\)-skew polynomial extensions
zbMATH Keywords: quantum Weyl algebras
reviewed by: T. H. Lenagan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 130638