Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T10:33:12.807Z Has data issue: false hasContentIssue false
  • English
  • Français

The structure of modules over polycyclic groups

Published online by Cambridge University Press:  24 October 2008

Kenneth A. Brown
Kenneth A. Brown
Affiliation:
University of Glasgow
Get access Rights & Permissions [Opens in a new window]

Abstract

The structure of modules over polycyclic group rings RG is investigated using the idea of a link P ⇝ Q between prime ideals of RG. The representation theory of RG splits into two parts – the part we discuss is determined by the representation theory of certain Noetherian polynomial identity factor rings of subrings of RG.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 2 , March 1981 , pp. 257 - 283
Copyright
Copyright © Cambridge Philosophical Society 1981

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.)

References

REFERENCES

(1)Brown, K. A.Module extensions over Noetherian rings. J. Algebra (to appear).Google Scholar
(2)Brown, K. A.Modules over polycyclic group rings have many irreducible images. Glasgow Math. J. (to appear 1981).CrossRefGoogle Scholar
(3)Brown, K. A.Artinian quotient rings of group rings. J. Algebra 49 (1977), 63–80.CrossRefGoogle Scholar
(4)Brown, K. A., Lenagan, T. H. and Stafford, J. T.K-theory and stable structure of some Noetherian group rings. Proc. London Math. Soc. (to appear).Google Scholar
(5)Burgess, W. D.Rings of quotients of group rings. Can. J. Math. 21 (1969), 865–975.CrossRefGoogle Scholar
(6)Cartan, H. and Eilenberg, S.Homological algebra (Princeton University Press, Princeton), 1956.Google Scholar
(7)Donkin, S. Locally finite representations of polycyclic-by-finite groups (to appear).Google Scholar
(8)Goldie, A. W. The structure of Noetherian rings. Lecture Notes in Mathematics no. 246 (Springer-Verlag, Berlin, 1972), 212–321.Google Scholar
(9)Gordon, R. and Robson, J. C.Krull dimension. Mem. Amer. Math. Soc. 133 (1973).Google Scholar
(10)Jategaonkar, A. W.Injective modules and localization in non-commutative Noetherian rings. Trans. Amer. Math. Soc. 190 (1974), 103–121.CrossRefGoogle Scholar
(11)Jategaonkar, A. W.Jacobson's conjecture and modules over fully bounded Noetherian rings. J. Algebra 30 (1974), 103–121.CrossRefGoogle Scholar
(12)Jategaonkar, A. W.Skew polynomial rings over orders in Artinian rings. J. Algebra 21 (1971), 51–59.CrossRefGoogle Scholar
(13)Krause, G., Lenagan, T. H. and Stafford, J. T.Ideal invariance and Artinian quotient rings. J. Algebra 55 (1978), 145–154.CrossRefGoogle Scholar
(14)Lorenz, M. and Passman, D. S. Prime ideals in group algebras of polycyclic-by-finite groups (to appear).Google Scholar
(15)Müller, B.Localisation in fully bounded Noetherian rings. Pac. J. Math. 67 (1976), 233–245.CrossRefGoogle Scholar
(16)Musson, I. M.Injective modules for group rings of polycyclic groups I. Quart. J. Math. (to appear).Google Scholar
(17)Passman, D. S.The Algebraic Structure of Group Rings (Interscience, 1977).Google Scholar
(18)Passman, D. S.On the ring of quotients of a group ring. Proc. Amer. Math. Soc. 33 (1972), 221–225.CrossRefGoogle Scholar
(19)Roseblade, J. E.Prime ideals in group rings of polycyclic groups. Proc. London Math. Soc. (3) 36 (1978), 385–447.CrossRefGoogle Scholar
(20)Roseblade, J. E. and Smith, P. F.A note on the Artin–Rees property of certain polycyclic group algebras. Bull. London Math. Soc. 11 (1979), 184–185.CrossRefGoogle Scholar
(21)Rosenberg, A. and Zelinsky, D.Finiteness of the injective hull. Math. Z. 70 (1959), 372–380.CrossRefGoogle Scholar
(22)Segal, D.On the residual simplicity of certain modules. Proc. London Math. Soc. 34 (1977), 327–353.CrossRefGoogle Scholar
(23)Smith, P. F.Quotient rings of group rings. J. London Math. Soc. (2) 3 (1971), 645–660.CrossRefGoogle Scholar
(24)Stenström, B.Rings of quotients (Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar