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Zero Divisors and Idempotents in Group Rings

Published online by Cambridge University Press:  20 November 2018

Gerald H. Cliff
Gerald H. Cliff*
Affiliation:
University of Alberta, Edmonton, Alberta
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We consider the following problem: If KG is the group ring of a torsion free group over a field K,show that KG has no divisors of zero. At characteristic zero, major progress was made by Brown [2], who solved the problem for G abelian-by-finite, and then by Farkas and Snider [4], who considered Gpolycyclic-by-finite. Here we present a solution at nonzero characteristic for polycyclic-by-finite groups. We also show that if Khas characteristic p > 0 and G is polycyclic-by-finite with only p-torsion, then KG has no idempotents other than 0 or 1. Finally we show that if R is a commutative ring of nonzero characteristic without nontrivial idempotents and G is polycyclic-by-finite such that no element different from 1 in G has order invertible in R, then RG has no nontrivial idempotents. This is proved at characteristic zero in [3].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 596 - 602
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bass, H., Euler characteristics and characters of discrete groups, Inv. Math. 35 (1976), 155–196.Google Scholar
2. Brown, K. A., On zero divisors in group rings, Bull. London Math. Soc. 8 (1976), 251–256.Google Scholar
3. Cliff, G. and Sehgal, S., On the trace of an idempotent in a group ring, Proc. Amer. Math. Soc. 62 (1977), 11–14.Google Scholar
4. Farkas, D. and Snider, R., Ko and Noetherian group rings, J. Algebr. 42 (1976), 192–198.Google Scholar
5. Formanek, E., Idempotents in Noetherian group rings, Can. J. Math. 15 (1973), 366–369.Google Scholar
6. Passman, D., The algebraic structure of group rings (Wiley-Interscience, New York, 1977).Google Scholar
7. Sehgal, S., Topics in group rings (Marcel Dekker, New York, 1978).Google Scholar
8. Serre, J.-P., Corps locaux (Hermann, Paris, 1962).Google Scholar
9. Zariski, O. and Samuel, P., Commutative algebra (Vol. I) (Van Nostrand, Princeton, 1958).Google Scholar