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Group rings which are v-HC orders and Krull orders

Published online by Cambridge University Press:  20 January 2009

K. A. Brown ,
H. Marubayashi  and
P. F. Smith
K. A. Brown
Affiliation:
Department of MathematicsUniversity of GlasgowUniversity Gardens Glasgow G12 8QW
H. Marubayashi
Affiliation:
Department of MathematicsUniversity of GlasgowUniversity Gardens Glasgow G12 8QW
P. F. Smith
Affiliation:
Department of MathematicsUniversity of GlasgowUniversity Gardens Glasgow G12 8QW
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Let R be a ring and G a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and G, when the group ring R[G] is a prime Krull order and when it is a price v-HC order. The key ingredient in obtaining both characterizations is the first author's earlier study of height one prime ideals in the ring R[G[.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 2 , June 1991 , pp. 217 - 228
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Brown, K. A., Height one primes of polycyclic group rings, J. London Math. Soc. (2) 32 (1985), 426–438.CrossRefGoogle Scholar
2.Brown, K. A., Hajarnavis, C. R. and MacEacharn, A. B., Rings of finite global dimension integral over their centres, Comm. Algebra 11 (1983), 67–93.CrossRefGoogle Scholar
3.Brown, K. A. and Hajarnavis, C. R., Homologically homogeneous rings, Trans. Amer. Math. Soc. 281 (1984), 197–208.CrossRefGoogle Scholar
4.Brown, K. A. and Hajarnavis, C. R., Injectively homogeneous rings, J. Pure Appl. Algebra 51 (1988), 65–77.CrossRefGoogle Scholar
5.Chamarie, M., Anneaux de Krull non commutatif, J. Algebra 72 (1981), 210–222.CrossRefGoogle Scholar
6.Lorenz, M. and Passman, D. S., Prime ideals in group algebras of polycyclic-by-finite groups, Proc. London Math. Soc. (3) 43 (1981), 520–543.CrossRefGoogle Scholar
7.Marubayashi, H., A Krull type generalization of HNP rings with enough invertible idelas, Comm. Algebra 11 (1983), 469–499.CrossRefGoogle Scholar
8.Marubayashi, H., A skew polynomial ring over a v-HC order with enough v-invertible ideals, Comm. Algebra 12 (1984), 1567–1593.CrossRefGoogle Scholar
9.Marubayashi, H., Remarks on VHC orders in a simple artinian ring, J. Pure Appl. Algebra 31 (1984), 109–118.CrossRefGoogle Scholar
10.Marubayashi, H., On v-ideals in a VHC order, Proc. Japan Acad. 59 (1983), 339–342.Google Scholar
11.Marubayashi, H., Divisorially graded rings by polycyclic-by-finite groups, Comm. Algebra 17 (1989), 2135–2177.CrossRefGoogle Scholar
12.Marubayashi, H., A note on divisorially graded rings by polycyclic-by-finite groups, Math. Japan. 34 (1989), 227–233.Google Scholar
13.Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano (Springer Lecture Notes in Math. 808, 1980).CrossRefGoogle Scholar
14.McConnell, J. C. and Robson, J. C., Non-commutative Noetherian Rings (Wiley, 1987).Google Scholar
15.Nastasescu, C., Nauwelaerts, E. and Van Oystaeyen, F., Arithmetically graded rings revisited, Comm. Algebra, to appear.Google Scholar
16.Passman, D. S., The algebraic structure of group rings (Wiley-Interscience, 1977).Google Scholar
17.Roseblade, J. E., Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3) 36 (1978), 385–447.CrossRefGoogle Scholar
18.Smith, P. F., Quotient rings of group rings, J. London Math. Soc. (2) 3 (1971), 645–660.CrossRefGoogle Scholar
19.Smith, P. F., Some example of maximal orders, Math. Proc. Cambridge Philos. Soc. 98 (1985), 19–32.CrossRefGoogle Scholar