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Localization in non-Noetherian group rings

Published online by Cambridge University Press:  18 May 2009

P. F. Smith
P. F. Smith
Affiliation:
Department of Mathematics, University Gardens, Glasgow G12 8QW
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Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p > 0, then kG is locally Noetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 21 , Issue 2 , July 1980 , pp. 151 - 163
Copyright
Copyright © Glasgow Mathematical Journal Trust 1980

References

REFERENCES

1.Brewer, J. W., Costa, D. L. and Lady, E. L., Prime ideals and localization in commutative group rings, J. Algebra 34 (1975), 300–308.CrossRefGoogle Scholar
2.Goldie, A. W., Semiprime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220.CrossRefGoogle Scholar
3.Herstein, I. N. and Small, L. W., Nil rings satisfying certain chain conditions, Canad. J. Math. 16 (1964), 771–776.CrossRefGoogle Scholar
4.Johnson, R. E. and Levy, L. S., Regular elements in semiprime rings, Proc. Amer. Math. Soc. 19 (1968), 961–963.CrossRefGoogle Scholar
5.Nouazé, Y. and Gabriel, P., Idéaux premiers de l'algèbre enveloppante d'une algèbre de Lie nilpotente, J. Algebra 6 (1967), 77–99.CrossRefGoogle Scholar
6.Passman, D. S., The algebraic structure of group rings (Wiley-Interscience, 1977).Google Scholar
7.Roseblade, J. E. and Smith, P. F., A note on hypercentral group rings, J. London Math. Soc. (2) 13 (1976), 183–190.CrossRefGoogle Scholar
8.Smith, P. F., Localization and the AR property, Proc. London Math. Soc. (3) 22 (1971), 39–68.CrossRefGoogle Scholar
9.Smith, P. F., On non-commutative regular local rings, Glasgow Math. J. 17 (1976), 98–102.CrossRefGoogle Scholar
10.Smith, P. F., The AR property and chain conditions in group rings, Israel J. Math. 32 (1979), 131–144.CrossRefGoogle Scholar
11.Smith, P. F., More on the AR property and chain conditions in group rings, Israel J. Math., to appear.Google Scholar
12.Walker, R., Local rings and normalizing sets of elements, Proc. London Math. Soc. (3) 24 (1972), 27–45.CrossRefGoogle Scholar