Maximal quotient rings of prime nonsingular group algebras
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
Recent work by Goodearl and Handelman has shown that a prime, regular, right self-injective ring Q must be precisely one of the following: (a) a full linear ring (being either simple Artinian; or infinite dimensional full linear) ; (b) a non-simple ring with zero socle ; (c) a directly finite, non-Artinian ring (necessarily simple) ; that is, an SP(∞) ring ; (d) a simple, directly infinite ring ; that is, an SP(1) ring which is not a division ring. Suppose now that Q is the maximal right quotient ring of a (necessarily prime nonsingular) group algebra KG. In this thesis we try to determine how this extra hypothesis affects the above classification. We prove in Chapter 1 that if all the conjugacy classes of G are countable then Q is either a full linear ring or a simple, directly infinite ring. In Chapter 2 we assume that G is locally finite. Using a dimension function on the finitely generated right ideals of KG, we show that if G is a nontrivial, locally finite group with only countable conjugacy classes then Q is simple and directly infinite. This is also true if G is the restricted symmetric group on any infinite set. In Chapter 3 we show that if Q is a full linear ring then G contains no nontrivial, locally finite, normal subgroups. If, in addition, G is soluble or residually finite or K has zero characteristic and G linear, then Q must be simple Artinian. In Chapter 4 we generalize the above-mentioned result from Chapter 1 and deduce that if G is soluble then Q must be a simple ring. Finally in Chapter 5 we study the ideals of Q and their interaction with the normal subgroups of G. We show that if G is residually finite then Q is a simple ring.