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Class groups and automorphism groups of group rings

Published online by Cambridge University Press:  18 May 2009

Kenneth A. Brown
Kenneth A. Brown
Affiliation:
Department of Mathematics, University of Glasgow, University Gardens, Glasgow, G12 8QW
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This paper is a sequel to [2]. A polycyclic-by-finite group G was there called dihedral free if G contains no subgroup isomorphic to 〈b, a:ba = b-1 a2 = 1〉 whose normalizer has finite index in G. It was shown in [2, Theorem F] that, if R is a commutative Noetherian domain, the group ring RG is a prime Noetherian maximal order if and only if R is integrally closed, G is dihedral free, and G has no non-trivial finite normal subgroups. Throughout, R and G will be assumed to satisfy these hypotheses. The main aim of the paper is to study the class group of the maximal order RG.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 79 - 86
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Bourbaki, N., Algèbre Commutative (Hermann, 1965).Google Scholar
2.Brown, K. A., Height one primes of polycyclic group rings, J. London Math. Soc., to appear.Google Scholar
3.Eisenbud, D., Subrings of Artinian and Noetherian rings, Math. Ann. 185 (1970), 247–249.CrossRefGoogle Scholar
4.Fossum, R. M., The Divisor Class Group of a Krull Domain (Springer-Verlag, 1970).Google Scholar
5.Goldman, O., Quasi-equality in maximal orders, J. Math. Soc. Japan 13 (1961), 371–375.CrossRefGoogle Scholar
6.Gruenberg, K. W., Cohomological Topics in Group Theory (Springer-Verlag, 1970).CrossRefGoogle Scholar
7.Bruyn, L. Le, Class groups of maximal orders over Krull domains, preprint (Antwerp, 1982).Google Scholar
8.Maury, G. and Raynaud, J., Ordres Maximaux au Sens de K. Asano (Springer-Verlag, 1980).CrossRefGoogle Scholar
9.Montgomery, S. and Passman, D. S., Crossed products over prime rings, Israel J. Math. 31 (1978), 224–256.CrossRefGoogle Scholar
10.Montgomery, S. and Passman, D. S., X-inner automorphisms of group rings, Houston J. Math. 7 (1981), 395–402.Google Scholar
11.Montgomery, S. and Passman, D. S., X-inner automorphisms of group rings II, Houston J. Math. 8 (1982), 537–544.Google Scholar
12.Passman, D. S., The Algebraic Structure of Group Rings (Wiley-Interscience, 1977).Google Scholar
13.Reiner, I., Maximal Orders (Academic Press, 1975).Google Scholar
14.Winter, D. J., The Structure of Fields (Springer-Verlag, 1974).CrossRefGoogle Scholar