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Zero divisors and idempotents in group rings

Published online by Cambridge University Press:  24 October 2008

P. A. Linnell
P. A. Linnell
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
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1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:

Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 3 , May 1977 , pp. 365 - 368
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Bass, H.Algebraic K-theory (New York, Benjamin, 1968).Google Scholar
(2)Brown, K. A.On zero divisors in group rings. Bull. London Math. Soc. 8 (1976), 251–256.CrossRefGoogle Scholar
(3)Casson, A. J. Whitehead groups of free products with amalgamation. In Algebraic K theory II, pp. 144–154 (Lecture Notes in Mathematics, no. 342, Berlin and New York, Springer-Verlag, 1973).Google Scholar
(4)Farkas, D. R. and Snider, R. L.K o and noetherian group rings. J. Algebra 42 (1976), 192–198.CrossRefGoogle Scholar
(5)Farrell, F. T. and Hsiano, W. C. A formula for K 1Rα[T], In Applications of Categorial Algebra, pp. 192–218 (Proceedings of the Symposium on Pure Mathematics, vol. 17, American Mathematical Society, Providence, R.I., 1970).CrossRefGoogle Scholar
(6)Formanek, E.Idempotents in noetherian group rings, Canari. J. Math. 15 (1973), 366–369.CrossRefGoogle Scholar
(7)Formanek, E.The zero divisor question for supersoluble groups. Bull. Austral. Math. Soc. 9 (1973), 69–71.CrossRefGoogle Scholar
(8)Hall, P.On the finiteness of certain soluble groups. Proc. London Math. Soc. (3) 9 (1959), 595–622.CrossRefGoogle Scholar
(9)Lewin, J.A note on zero divisors in group rings. Proc. Amer. Math. Soc. 31 (1972), 357–359.CrossRefGoogle Scholar
(10)Passman, D. S.Advances in group rings. Israel J. Math. 19 (1974), 67–107.CrossRefGoogle Scholar
(11)Swan, R.Induced representations and projective modules. Ann. of Math. 71 (1960), 552–578.CrossRefGoogle Scholar