Speaker: Warren McGovern, Florida Atlantic University
Title: Spaces of Maximal Ideals and Z-Ultrafilters
Abstract: Let R be a commutative ring with identity. Denote the set of maximal ideals of R by Max(R). Given an ideal I\subseteq R, define
$V(I)={M\in Max(R) : I\subseteq M}$
The collection Z={V(I) : I is a f.g. ideal} is closed under finite unions and finite intersections. Thus, it is a bounded distributive lattice of sets. Moreover, Z is a Wallman lattice. Therefore, the collection of Z-ultrafilters, Ult(Z), is a compact T_1 space when equipped with the Wallman topology. There is a one-to-one correspondence between the collection of z-ideals of R and the collection of Z-filters. In particular, Max(R) and Ult(Z) are homeomorphic. This produces a new way of characterizing when Max(R) is Hausdorff.
Our goal is to generalize this to the lattice Z^# = {cl int V(I) : I is f.g. ideal}. In particular, we would like to describe Ult (Z^#) as well as the corresponding maximal Z^# ideals.