Spaces X in which all prime z-ideals of C(X) are minimal or maximal
by
Melvin Henriksen
Harvey Mudd College Claremont, CA 91711

X will denote a Tychonoff space and C(X)the ring of continuous real-valued functions on X. Suzanne Larson calls C(X) quasinormal if the sum of two distinct minimal prime ideals of C(X) is either a maximal ideal or all of C(X), and X is called a quasi P-space if all prime z-ideals of C(X) are minimal or maximal. The rank of a maximal ideal is the number of minimal prime ideals contained in it if this is a positive integer, and infinity otherwise.

Main Theorem: Suppose X is compact.    (a)C(X) is quasinormal iff the set of points of X of rank 1 is cofinite and each such point has a compact neighborhood that is an F-space in the sense of Gillman and Jerison, and    (b) X is a quasi P-space iff it is scattered of Cantor-Bendixson order 1 or 2.

What happens when X is not compact is discussed. This is part of joint research with Jorge Martinez and R. G. Woods.

Date received: May 3, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-16.