A Characterization of Biregular Group Rings

Abstract

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]).