Commutative group rings with von Neumann regular total rings of quotients

Abstract

Let R be a commutative ring and let G be an abelian group. We show that if G is either torsion free or R is uniquely divisible by the order of every element of G, then the von Neumann regularity of the total ring of quotients of R ascends to the total ring of quotients of RG. Examples are given to show that the converse does not hold. These results are applied in the group ring setting to explore a number of zero divisor controlling conditions, such as being a PF or a PP ring as well as a number of Prüfer conditions, such as being an arithmetical, a Gaussian, or a Prüfer ring.