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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
2. Burgess, W. D. and Stephenson, W., Pierce sheaves of non-commutative rings, Comm. Algebra, 4 (1976), 51-75.Google Scholar
3. Jacobson, N., Structure of Rings. Amer. Math. Soc. Colloquium Publications, 37, Providence, R.I., 1964.Google Scholar
4. Passman, D. S., Infinite Group Rings. Marcel Dekker Inc. New York, 1971.Google Scholar
5. Pierce, R. S., Modules over commutative regular rings, Mem. of the Amer. Math. Soc, 70 (1967).Google Scholar
6. Renault, G., Anneaux biréguliers auto-injectifs à droite. J. Algebra, 36 (1975), 77-84.Google Scholar
7. Renault, G., Anneaux de groupes biréguliers. Séminaire d'algèbre non-commutative, 1973, Publications mathématiques d'orsay.Google Scholar