Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-31T08:35:10.116Z Has data issue: false hasContentIssue false
  • English
  • Français

Affine algebras of Gelfand-Kirillov dimension one are PI

Published online by Cambridge University Press:  24 October 2008

L. W. Small ,
J. T. Stafford  and
R. B. Warfield Jr
L. W. Small
Affiliation:
Department of Mathematics, UCSD, La Jolla, CA 92093, U.S.A.
J. T. Stafford
Affiliation:
Department of Pure Mathematics, Leeds University, Leeds LS2 9JT, England
R. B. Warfield Jr
Affiliation:
Department of Mathematics, University of Washington, Seattle, W A 98195, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

The aim of this paper is to prove:

Theorem. Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 407 - 414
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Borho, W. and Kraft, H., Über die Gelfand-Kirillov-Dimension. Math. Ann. 220 (1976), 1–24.CrossRefGoogle Scholar
[2]Braun, A.. A note on Noetherian PI rings. Proc. Amer. Math. Soc. 83 (1981), 670–672.CrossRefGoogle Scholar
[3]Goldie, A. W.. Semiprime rings with minimum condition. Proc. London Math. Soc. 10 (1960), 201–220.CrossRefGoogle Scholar
[4]Herstein, I. N.. Noncommutative Rings. Carus Monograph, no. 15. Math. Soc. of Amer., 1968.Google Scholar
[5]Irving, R. S.. Affine algebras with any set of integers as the dimensions of simple modules. Bull. London Math. Soc. (In the Press).Google Scholar
[6]Irving, R. S. and Small, L. W.. The Goldie conditions for algebras of bounded growth. Bull. London Math. Soc. 15 (1983), 596–600.CrossRefGoogle Scholar
[7]Krause, G. and Lenagan, T. H.. Growth of Algebras and Gelfand-Kirillov Dimension. Research Notes in Math. Vol. 116 (Pitman, 1985).Google Scholar
[8]Lewin, J.. Subrings of finite index in finitely generated rings. J. Algebra 5 (1967), 84–88.CrossRefGoogle Scholar
[9]Macaulay, F. S.. Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc. 56 (1927), 531–555.CrossRefGoogle Scholar
[10]Procesi, C.. Rings with Polynomial Identity (Dekker, 1973).Google Scholar
[11]Small, L. W.. An example in PI rings. J. Algebra 17 (1971), 434–436.CrossRefGoogle Scholar
[12]Small, L. W. and Warfield, R. B.. Prime affine algebras of Gelfand-Kirillov dimension one. J. Algebra 91 (1984), 386–389.CrossRefGoogle Scholar