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Commutative Algebra Seminar -- Warren McGovern

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January 23, 2017
4:15PM - 5:15PM
Cockins Hall 240

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Add to Calendar 2017-01-23 16:15:00 2017-01-23 17:15:00 Commutative Algebra Seminar -- Warren McGovern Speaker:  Warren McGovern, Florida Atlantic UniversityTitle:  Spaces of Maximal Ideals and Z-UltrafiltersAbstract:  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. Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public

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.

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