Commutative Rings whose Prime Spectra have certain Arithmetical Closure Properties
In this thesis, we consider two questions. First, let R be a commutative ring with identity. Suppose the collection of prime ideals of R is closed under ideal multiplication. What does this condition imply about the ideal structure of R? Dually, suppose instead that the collection of nonprime ideals of R is closed under addition. Again, how does this condition influence the ideal structure of R? We give partial answers to these questions, and also indicate potential directions for deeper investigations.