Zero-Divisor Conditions in Commutative Group Rings

Date of Completion

January 2011






Let R be a commutative ring and let G be an abelian group. Basic ways to control zero-divisors in a commutative group ring RG are to require it to be a domain or less restrictively, to be a reduced ring. Higman [31] found necessary and sufficient conditions for group rings to be integral domains, while May [43] characterized reduced group rings. Further zero-divisor controlling conditions include the following: (1) R is a PF ring, i.e. every principal ideal of R is flat. (2) R is a PP ring, i.e. every principal ideal of R is projective. (3) Q( R), total ring of quotients of R, is von Neumann regular. (4) Min R, the set of minimal primes of R, is compact in the Zariski topology.^ In this dissertation, we examine the ascent and descent of these zero-divisor controlling conditions between R and RG, where G is either a torsion free group or R is uniquely divisible by all prime orders of elements of G. Examples of group rings exhibiting these conditions are also given. As an application, connections between these zero-divisor controlling conditions and Priifer conditions in RG are investigated. ^