Geodesic
A Characterization of Biregular Group Rings
Canadian mathematical bulletin, Tome 21 (1978) no. 1, pp. 119-120

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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]). 
Burgess, W. D. A Characterization of Biregular Group Rings. Canadian mathematical bulletin, Tome 21 (1978) no. 1, pp. 119-120. doi: 10.4153/CMB-1978-020-8
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[1] 1. Bovdi, A. and Mihovski, S. V., Idempotents in crossed products, Sov. Math. Dokl. 11 (1970), 1439-1441. Google Scholar

[2] 2. Burgess, W. D. and Stephenson, W., Pierce sheaves of non-commutative rings, Comm. Algebra, 4 (1976), 51-75. Google Scholar

[3] 3. Jacobson, N., Structure of Rings. Amer. Math. Soc. Colloquium Publications, 37, Providence, R.I., 1964. Google Scholar

[4] 4. Passman, D. S., Infinite Group Rings. Marcel Dekker Inc. New York, 1971. Google Scholar

[5] 5. Pierce, R. S., Modules over commutative regular rings, Mem. of the Amer. Math. Soc, 70 (1967). Google Scholar

[6] 6. Renault, G., Anneaux biréguliers auto-injectifs à droite. J. Algebra, 36 (1975), 77-84. Google Scholar

[7] 7. Renault, G., Anneaux de groupes biréguliers. Séminaire d'algèbre non-commutative, 1973, Publications mathématiques d'orsay. Google Scholar

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