Abstract
In this paper, we prove that certain group rings are primitive by exhibiting a construction of faithful, simple modules for them. Let us recall two of the early results on primitivity of group rings. Let G be an infinite group, and let Z 2 G be the wreath product of the infinite cyclic groupZ with G. Passman proved that for fields K with transcendence degree greater than or equal to the cardinality of G, the group ring K[Z 2 G] is primitive [8]. For any other group H, Formanek proved that the group ring of the free product G*H is primitive over any field, or any domain of cardinality less than or equal to the cardinality of G [3]. Both proofs show that a comaximal left ideal exists, in order to deduce primitivity. More recently, McGregor has exhibited a faithful, simple module for the group ring of a countable free group over a field [8]. W e p rove in Section 2 an extension of Passman’s result:
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