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”.


16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
16P50 Localization and associative Noetherian rings
Full Text: DOI


[1] Amitsur, S. A., A generalization of a theorem on linear differential equations, Bull. Amer. Math. Soc., 54, 937-941 (1948) · Zbl 0034.19803
[2] Andrews, G. E., The theory of partitions, (Rota, G.-C., Encyclopedia of Math. and Its Applic., Vol. 2 (1976), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0155.09302
[3] Awami, M.; Van den Bergh, M.; Van Oystaeyen, F., Note on derivations of graded rings and classification of differential polynomial rings, Bull. Soc. Math. Belg. Sér. A, 40, 175-183 (1988) · Zbl 0663.16001
[4] Bell, A. D., Localization and Ideal Theory in Noetherian Crossed Products and Differential Operator Rings, (Ph.D. Dissertation (1984), University of Washington)
[5] Bell, A. D., When are all prime ideals in an Ore extension Goldie?, Comm. Algebra, 13, 1743-1762 (1985) · Zbl 0567.16002
[6] Bell, A. D., Localization and ideal theory in Noetherian strongly group-graded rings, J. Algebra, 105, 76-115 (1987) · Zbl 0607.16013
[7] Bell, A. D., Localization and ideal theory in iterated differential operator rings, J. Algebra, 106, 376-402 (1987) · Zbl 0608.16033
[8] Bell, A. D., Notes on localization in noncommutative Noetherian rings, (Cuadernos de Algebra, Vol. 9 (1988), Universidad de Granada: Universidad de Granada Granada)
[9] Bell, A. D.; Musson, I. M., Primitive factors of enveloping algebras of nilpotent Lie superalgebras, J. London Math. Soc., 42, 401-408 (1990) · Zbl 0682.17005
[10] Brown, K. A.; Warfield, R. B., Krull and global dimensions of fully bounded noetherian rings, (Proc. Amer. Math. Soc., 92 (1984)), 169-174 · Zbl 0557.16009
[11] Brown, K. A.; Warfield, R. B., The influence of ideal structure on representation theory, J. Algebra, 116, 294-315 (1988) · Zbl 0652.16006
[12] Cohn, P. M., Quadratic extensions of skew fields, Proc. London Math. Soc., 11, 3, 531-556 (1961) · Zbl 0104.03301
[13] Cohn, P. M., (Algebra, Vol. 2 (1977), Wiley: Wiley London)
[14] Drinfeld, V. G., Quantum groups, (Proceedings, Internat. Congr. Math., Berkeley, Vol. 1 (1986)), 798-820
[16] Fisher, J. R., A Goldie theorem for differentiably prime rings, Pacific J. Math., 58, 71-77 (1975) · Zbl 0311.16036
[17] Goldie, A. W.; Michler, G., Ore extensions and polycyclic group rings, J. London Math. Soc., 9, 2, 337-345 (1974) · Zbl 0294.16019
[18] Goodearl, K. R., Global dimension of differential operator rings, (Proc. Amer. Math. Soc., 45 (1974)), 315-322 · Zbl 0263.13003
[20] Goodearl, K. R.; Warfield, R. B., Primitivity in differential operator rings, Math. Z., 180, 503-523 (1982) · Zbl 0495.16002
[21] Goodearl, K. R.; Warfield, R. B., An introduction to noncommutative Noetherian rings, (London Math. Soc. Student Text Series, Vol. 16 (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 1101.16001
[22] Hayashi, T., \(Q\)-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys., 127, 129-144 (1990) · Zbl 0701.17008
[23] Hodges, T. J., Ring-theoretical aspects of the Bernstein-Beilinson theorem, (Noncommutative Ring Theory (Athens, Ohio, 1989). Noncommutative Ring Theory (Athens, Ohio, 1989), Lecture Notes in Math., Vol. 1448 (1990), Springer-Verlag: Springer-Verlag Berlin), 155-163
[24] Irving, R. S., Prime ideals of Ore extensions over commutative rings, J. Algebra, 56, 315-342 (1979) · Zbl 0399.16015
[25] Irving, R. S., Prime ideals of Ore extensions over commutative rings, II, J. Algebra, 58, 399-423 (1979) · Zbl 0411.16025
[26] Jackson, F. H., Generalization of the differential operative symbol with an extended form of Boole’s equation \(ϱ(ϱ\)−1)\((ϱ\)−2)…(ϱ−n+1)=\(x^n(d^n/dx^{nn} )\), Messenger Math., 38, 57-61 (1908-1909)
[27] Jategaonkar, A. V., Skew polynomial rings over semisimple rings, J. Algebra, 19, 315-328 (1971) · Zbl 0223.16005
[28] Jategaonkar, A. V., Localization in Noetherian rings, (London Math. Soc. Lecture Notes, Vol. 98 (1986), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0589.16014
[29] Jimbo, M., A \(q\)-difference analog of \(U\)(g) and the Yang-Baxter equation, Lett. Math. Phys., 10, 63-69 (1985) · Zbl 0587.17004
[30] Jordan, D. A., Ore Extensions and Jacobson Rings, (Ph.D. Dissertation (1975), University of Leeds) · Zbl 0313.16011
[31] Jordan, D. A., Noetherian Ore extensions and Jacobson rings, J. London Math. Soc., 10, 281-291 (1975) · Zbl 0313.16011
[34] Kuryshkin, V., Opérateurs quantiques généralisés de création et d’annihilation, Ann. Fondation L. Broglie, 5, 111-125 (1980)
[35] Lenagan, T. H.; Warfield, R. B., Affiliated series and extensions of modules, J. Algebra, 142, 164-187 (1991) · Zbl 0743.16016
[36] Malm, D. R., Simplicity of partial and Schmidt differential operator rings, Pacific J. Math., 132, 85-112 (1988) · Zbl 0608.16005
[37] McConnell, J. C.; Robson, J. C., (Noncommutative Noetherian Rings (1987), Wiley-Interscience: Wiley-Interscience New York) · Zbl 0644.16008
[38] Montgomery, S.; Smith, S. P., Skew derivations and \(U_q\)(sl(2)), Israel J. Math., 72, 158-166 (1990) · Zbl 0717.16031
[39] Morikawa, H., On \(ξ_n\)-Weyl algebra \(W_r(ξ_n\), Z), Nagoya Math. J., 113, 153-159 (1989)
[40] Pearson, K. R.; Stephenson, W., A skew polynomial ring over a Jacobson ring need not be a Jacobson ring, Comm. Algebra, 5, 783-794 (1977) · Zbl 0355.16020
[41] Posner, E. C., Prime rings satisfying a polynomial identity, (Proc. Amer. Math. Soc., 11 (1960)), 180-183 · Zbl 0215.38101
[42] Resco, R.; Small, L. W.; Stafford, J. T., Krull and global dimensions of semiprime noetherian PI-rings, Trans. Amer. Math. Soc., 274, 285-295 (1982) · Zbl 0495.16008
[43] Rowen, L. H., (Ring Theory, Vol. II (1988), Academic Press: Academic Press San Diego)
[44] Sigurdsson, G., Links between prime ideals in differential operator rings, J. Algebra, 102, 260-283 (1986) · Zbl 0597.16004
[45] Stafford, J. T., The Goldie rank of a module, (Small, L. W., Noetherian Rings and Their Applications. Noetherian Rings and Their Applications, Math Surveys and Monographs, Vol. 24 (1987), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-20 · Zbl 0632.16014
[46] Stanley, R. P., (Enumerative Combinatorics, Vol. I (1986), Wadsworth & Brooks/Cole: Wadsworth & Brooks/Cole Monterey, CA) · Zbl 0608.05001
[47] Warfield, R. B., Noncommutative localized rings, (Sém. d’Alg. P. Dubreil et M.-P. Malliavin. Sém. d’Alg. P. Dubreil et M.-P. Malliavin, Lecture Notes in Math., Vol. 1220 (1986), Springer-Verlag: Springer-Verlag Berlin), 178-200 · Zbl 0602.16015
[48] Wexler-Kreindler, E., Propriétés de transfert des extensions d’Ore, (Sém. d’Alg. P. Dubreil, 1976-1977. Sém. d’Alg. P. Dubreil, 1976-1977, Lecture Notes in Math., Vol. 641 (1978), Springer-Verlag: Springer-Verlag Berlin), 235-251 · Zbl 0375.16029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.