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Abstract
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.
