Cardinalities of Residue Fields of Noetherian Integral Domains

Abstract

We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in Zermelo–Fraenkel Set Theory with Choice (ZFC) that there is a Noetherian domain of cardinality ℵ₁ with a finite residue field, but the statement “There is a Noetherian domain of cardinality ℵ₂ with a finite residue field” is equivalent to the negation of the Continuum Hypothesis.

Key Words:

• Generalized continuum hypothesis
• Integral domain
• Noetherian ring
• Prime spectrum
• Residue field

2000 Mathematics Subject Classification:

• Primary 13E05
• Secondary 13B25, 13H99

Notes

Communicated by I. Swanson.