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

X will denote a Tychonoff space and C(X) the ring of continuous real-valued functions on X. Notation and terminology used is that of Gillman and Jerison's Rings of continuous functions. If every prime z-deal of C(X) is minimal or maximal, then X is called a quasi P-space. 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).

SAMPLE RESULTS:

• Every space with only finitely many non P-points is quasiP, but there is a quasi P-space with no P-points.
• Every F-space is quasinormal, every quasinormal space is a quasi P-space, but no quasi P-space can have a compact F-space as a subspace.
• A locally compact space is a quasi P-space iff it is scattered of Cantor-Bendixon index 1 or 2, and every compact quasi P-space is a finite free union of finitely many spaces of countable ordinals.
• The product of a compact quasi P-space and a P-space is a quasi P-space.

This is part of joint research with Jorge Martinez and R. G. Woods. We have also studied spaces X such that every chain of prime z-ideals is finite.

Date received: July 11, 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 # cagx-34.