Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-07-05T09:53:11.317Z Has data issue: false hasContentIssue false
  • English
  • Français

Some examples of noncommutative local rings

Published online by Cambridge University Press:  18 May 2009

D. A. Jordan
D. A. Jordan
Affiliation:
Department of Pure MathematicsUniversity of SheffieldHicks Building Hounsfield Road Sheffield S3 7RH
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we construct examples which answer three questions in the general area of noncommutative Noetherian local rings and rings of finite global dimension. The examples are formed in the same basic way, beginning with a commutative polynomial ring A over a field k and a k-derivation δ of A, taking the skew polynomial ring R = A[x;δ] and localizing at a prime ideal of the form IR, where I is a prime ideal of A invariant under δ. The localization is possible by a result of Sigurdsson [13].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 1 , January 1990 , pp. 79 - 86
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Bell, A. D. and Sigurdsson, G., Catenarity and Gelfand-Kirillov dimension in Ore extensions, preprint (University of Southern California-Northern Illinois University).Google Scholar
2.Brown, K. A., Hajarnavis, C. R. and MacEacharn, A. B., Noetherian rings of finite global dimension, Proc. London Math. Soc. 44 (1982), 349–371.CrossRefGoogle Scholar
3.Chatters, A. W. and Jordan, D. A., Non-commutative unique factorization rings, J. London Math. Soc. (2) 33 (1986), 22–32.CrossRefGoogle Scholar
4.Cohn, P. M., Non-commutative unique factorisation domains, Trans Amer. Math. Soc. 109 (1963), 313–331.Google Scholar
5.Hajarnavis, C. R. and Williams, S., Maximal orders in Artinian rings, J. Algebra 90 (1984), 375–384.Google Scholar
6.Jordan, D. A., Noetherian Ore extensions and Jacobson rings, J. London Math. Soc. (2) 10 (1975), 281–291.Google Scholar
7.Jordan, D. A., Primitive Ore extensions, Glasgow Math. J. 18 (1977), 93–97.CrossRefGoogle Scholar
8.Kaplansky, I., Commutative rings, revised edition (University of Chicago Press, 1974).Google Scholar
9.Maury, G., Nouveaux exemples d'ordres maximaux, Comm. Algebra 14 (1986), 1515–1517.CrossRefGoogle Scholar
10.Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano, Lecture Notes in Mathematics 808 (Springer, 1980).Google Scholar
11.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings, (Wiley, 1988).Google Scholar
12.Müller, B. J., Localization in non-commutative Noetherian rings, Canad. J. Math. 28 (1976), 600–610.Google Scholar
13.Sigurdsson, G., Links between prime ideals in differential operator rings, J. Algebra 102 (1986), 260–283.Google Scholar
14.Walker, R., Local rings and normalizing sets of elements, Proc. London Math. Soc. (3) 24 (1972), 27–45.Google Scholar