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Prime ideals in skew Laurent polynomial rings

Published online by Cambridge University Press:  20 January 2009

K. W. Mackenzie
K. W. Mackenzie
Affiliation:
Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ
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Abstract

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Let R be a commutative ring and {σ1,…,σn} a set of commuting automorphisms of R. Let T = be the skew Laurent polynomial ring in n indeterminates over R and let be the Laurent polynomial ring in n central indeterminates over R. There is an isomorphism φ of right R-modules between T and S given by φ(θj) = xj. We will show that the map φ induces a bijection between the prime ideals of T and the Γ-prime ideals of S, where Γ is a certain set of endomorphisms of the ℤ-module S. We can study the structure of the lattice of Γ-prime ideals of the ring S by using commutative algebra, and this allows us to deduce results about the prime ideal structure of the ring T. As an example, if R is a Cohen-Macaulay ℂ-algebra and the action of the σj on R is locally finite-dimensional, we will show that the ring T is catenary.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 2 , June 1993 , pp. 299 - 317
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Atiyah, M. F. and McDonald, I. G., Introduction to Commutative Algebra (Addison-Wesley, Reading, Mass., 1969).Google Scholar
2.Bell, A. D. and Sigurdsson, G., Catenarity and Gelfand–Kirillov dimension in Ore extensions, J. Algebra 127 (1989), 409–425.CrossRefGoogle Scholar
3.Brown, K. A., Goodearl, K. R. and Lenagan, T. H., Prime ideals in differential operator rings—catenarity, Trans. Amer. Math. Soc. 317 (1990), 749–772.CrossRefGoogle Scholar
4.Humphreys, J. E., Linear Algebraic Groups (Graduate Texts in Mathematics 21, Springer-Verlag, New York-Heidelberg-Berlin, 1981).Google Scholar
5.Jategaonkar, A. V., Localization in Noetherian Rings (London Math. Soc. Lecture Note Series 98, Cambridge University Press, Cambridge, 1986).CrossRefGoogle Scholar
6.Kunz, E., Introduction to Commutative Algebra and Algebraic Geometry (Birkhäuser, Boston, 1985).Google Scholar
7.Matsumura, H., Commutative Ring Theory (Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986).Google Scholar
8.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian Rings (Wiley series in Pure and Applied Mathematics, Wiley, 1987).Google Scholar
9.Roseblade, J. E., Prime ideals in group rings of polycyclic groups, Proc. Lond. Math. Soc. (3) 36 (1978), 385–447; Corrigenda: Prime ideals in group rings of polycyclic groups. Proc. London Math. Soc (3) 38 (1979), 216–18.CrossRefGoogle Scholar