#### Title

Zero-Divisor Conditions in Commutative Group Rings

#### Date of Completion

January 2011

#### Keywords

Mathematics|Nanotechnology

#### Degree

Ph.D.

#### 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. ^

#### Recommended Citation

Schwarz, Ryan Peter, "Zero-Divisor Conditions in Commutative Group Rings" (2011). *Doctoral Dissertations*. AAI3492156.

https://opencommons.uconn.edu/dissertations/AAI3492156