A UNIVERSAL SURVIVAL RING OF CONTINUOUS FUNCTIONS WHICH IS NOT A UNIVERSAL LYING-OVER RING

A UNIVERSAL SURVIVAL RING OF CONTINUOUS FUNCTIONS WHICH IS NOT A UNIVERSAL LYING-OVER RING

The ring R of continuous real-valued functions on the one-point compactification of the discrete space of cardinality ℵ₁ is a universal survival ring, yet is not a ULO-ring. Chains of prime ideals of R of cardinality c exist. Moreover, R/P is a divided domain for each P ϵ Spec (R). If the Continuum...

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Veröffentlicht in:The Rocky Mountain journal of mathematics 2013-01, Vol.43 (3), p.825-854
Hauptverfasser: DOBBS, DAVID E., LEVY, RONALD, SHAPIRO, JAY
Format: Artikel
Sprache:eng
Schlagworte:
13A15
13B99
13F05
54C35
54D35
continuum hypothesis
divided domain
Krull dimension
lying-over
one-point compactification
prime ideal
ring extension
ring of continuous functions
ultrafilter
Universal survival ring
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Zusammenfassung:The ring R of continuous real-valued functions on the one-point compactification of the discrete space of cardinality ℵ₁ is a universal survival ring, yet is not a ULO-ring. Chains of prime ideals of R of cardinality c exist. Moreover, R/P is a divided domain for each P ϵ Spec (R). If the Continuum Hypothesis holds, then there exists a minimal prime ideal P of R such that R/P is an infinite-dimensional valuation domain; however, it is consistent with ZFC that no such minimal primes exist.
ISSN:0035-7596
1945-3795
DOI:10.1216/RMJ-2013-43-3-825