Maximal Quotient Rings of Prime Group Algebras

Abstract

Abstract. It is shown that a prime nonsingular group algebra KG, where G is a group whose conjugacy classes are countable, has for its maximal right quotient ring either a full linear ring or a simple directly infinite ring. In the case where G is also locally finite only the second type can occur. 0. Introduction. The maximal right quotient ring Q of a prime nonsingular ring A* is a prime regular right self-injective ring. Accordingly, within the class of such rings, Q falls into one of the following disjoint subclasses (see Goodearl and Handelman [7]):(a) full linear rings, (b) nonsimple rings with zero socle, (c) directly finite non-Artinian rings (necessarily simple), (d) simple, directly infinite rings. All four types are possible for a general R (see [7] for examples of (c) and (d); for (b) take R = T/soc T where T = Homf (F, V) for some vector space V over a countable field F with dimF V > 2"°). However in § 1 we show that if A is a prime nonsingular group algebra AG, where G is a group whose conjugacy classes are countable, then only (a) and (d) can occur; type (a) if AG has uniform right ideals, type (d) otherwise. This result is deduced after the corresponding classification is established when A is a countable-dimensional algebra. In §2 we show that uniform right ideals do not exist in AG if G is also locally finite, and so the maximal quotient ring in this case is always simple and directly infinite.The above results are in direct contrast with Kaplansky's result that group algebras over a field of characteristic zero are directly finite. Should some class of group algebras have directly finite maximal quotient rings, this would of course also establish the direct finiteness of these group algebras (and their subgroup-algebras).Note. Maximal right quotient ring (MRQ ring) is used in the sense of R. E. Johnson [10]. A nonsingular ring is one with zero right singular ideal. Such rings always have an MRQ ring, which is regular and right self-injective. For modules A and B we write A < B if A is isomorphic to a submodule of B. A ring is directly finite if all one-sided inverses are two-sided (i.e. xy = 1 implies