Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T20:16:38.440Z Has data issue: false hasContentIssue false

Ideals in group rings of soluble groups of finite rank

Published online by Cambridge University Press:  24 October 2008

Christopher J. B. Brookes
Affiliation:
Corpus Christi College, Cambridge

Extract

The original motivation for this paper was the question of primitivity for group rings of soluble groups of finite rank. At the end this is touched upon as an application of a theorem about prime ideals in such rings. If a group Γ acts on a set S we say an element is (Γ)-orbital if its orbit is finite and write ΔΓ(S) for the subset of such elements. The FC-radical of a group G, denoted by Δ(G), is just ΔG(G) where the action of G on itself is by conjugation.

Type
Research Article
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.)

References

REFERENCES

[1]Bergman, G. M.The logarithmic limit-set of an algebraic variety. Trans. Amer Math, Soc, 157 (1971), 459469Google Scholar
[2]Bieri, R. and Strebel, R.Almost finitely presented soluble groups. Comment. Math. Helv. 53 (1978), 258278.Google Scholar
[3]Bieri, R. and Strebel, R.Soluble Groups with Coherent Group Rings. London Math. Soc. Lecture. Notes no. 36 (Cambridge University Press, 1979), 235240.Google Scholar
[4]Bieri, R. and Strebel, R.Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41 (1980), 439464.Google Scholar
[5]Brewster, D. C. The maximum condition on ideals of the group ring. Ph.D. dissertation, Cambridge University, 1976.Google Scholar
[6]Gruenberg, K.Ring Theoretic Methods and Finiteness Conditions in Infinite Soluble Group Theory. Lecture Notes in Mathematics no. 319 (Springer-Verlag, Berlin, 1973), 7584.Google Scholar
[7]Musson, I. M.Representations of infinite soluble groups. Glasgow Math. J. 24 (1983), 4352.Google Scholar
[8]Passman, D. S.The Algebraic Structure of Group Rings (Wiley Interscience, 1977).Google Scholar
[9]Roseblade, J. E.Prime ideals in group rings of polycyclic groups. Proc. London Math. Soc. (3) 36 (1978), 385447.Google Scholar
[10]Wehrfritz, B. A. F.Infinite Linear Groups (Springer-Verlag, 1973).CrossRefGoogle Scholar
[11]Zareski, A. E. Intersection theorems in group rings. (In Russian.) VINITI (Moscow, 1974).Google Scholar
[12]Zariski, O. and Samuel, P.Commutative Algebra, vol. I and II. (Van Nostrand, 1958 and 1960).Google Scholar