Aspects of the zero divisor problem

The Zero Divisor Problem is the following:- lf G is a torsion-free group and R is a commutative domain, is RG a domain? This thesis is concerned with three aspects of this problem. After stating various background results in Chapter 1, we prove in Chapter 2 that RG is a domain if R is a commutative domain of characteristic zero, and G is a torsion-free group which is in one of various classes of groups, of which the most important is the class of abelian-by-finite groups. If R is a commutative ring and G is a soluble group such that RG is a domain, then RG is an Ore domain, and so has a division ring of quotients. We are thus led in Chapter 3 to investigate under what circumstances group rings of generalised soluble groups have Artinian quotient rings. We obtain results for a class of groups which includes many (but not all) torsion-free soluble groups, and we show that, with appropriate assumptions on the coefficient rings, the quotient rings in question are QF-rings. Chapter 3 also includes several applications and examples. In Chapter 4 we study the zero divisors of group rings by investigating the structure of the singular ideals. We explicitly describe the singular ideals of the group algebras of various classes of groups, (including soluble groups). The results obtained are reminiscent of results of Passman, Zalesskii et al on the structure of the Jacobson radical. We include various examples and applications. Each chapter begins with a detailed introduction