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A Characterization of Biregular Group Rings

Published online by Cambridge University Press:  20 November 2018

W. D. Burgess
W. D. Burgess*
Affiliation:
University of Ottawa, Ottawa, Ontario
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In this note biregular group rings are characterized and an example is given to show that Renault′s conjecture is false.

A ring A with 1 is biregular if for all a∈A, AaA is generated by a central idempotent Equivalently, A is biregular iff all the stalks of its Pierce sheaf are simple.

In [1] Bovdi and Mihovski showed that for a ring A, if the group ring AG is biregular then: (*) A is biregular and G is locally normal with the order of each finite normal sub-group of G invertible in A. A proof is found in Renault [7]. In [6] Renault showed that (*) is necessary and sufficient in case A is a finitely generated module over its centre or if A is right self-injective. He conjectured that (*) is necessary and sufficient in general. In fact (*) is not sufficient as the example below shows. Some familiarity with Pierce sheaf techniques is assumed (see [5] or [2]).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 1 , 01 March 1978 , pp. 119 - 120
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Bovdi, A. and Mihovski, S. V., Idempotents in crossed products, Sov. Math. Dokl. 11 (1970), 1439-1441.Google Scholar
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4. Passman, D. S., Infinite Group Rings. Marcel Dekker Inc. New York, 1971.Google Scholar
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7. Renault, G., Anneaux de groupes biréguliers. Séminaire d'algèbre non-commutative, 1973, Publications mathématiques d'orsay.Google Scholar