Prime elements from prime ideals
type: Item
instance of: scholarly article
title: Prime elements from prime ideals
zbMATH Open document ID: 0576.06010
DOI: 10.1007/BF00334858
published in: Order
publication date: 1985-01-01T00:00:00Z
review text: In the summer of 1984 A. R. Blass proved that the ''Almost Maximal Ideal Theorem'' which had been introduced by the reviewer [Fundam. Math. 123, 197-209 (1984; Zbl 0552.06004)] - and which he had previously believed to be a choice principle intermediate between the Prime Ideal Theorem and the Axiom of Choice - was in fact logically equivalent to the Prime Ideal Theorem. Blass' proof was a slightly involved one making use of the compactness theorem for first-order logic; on receiving news of it, both B. Banaschewski and the reviewer independently discovered simpler proofs. Here is Banaschewski's proof, which goes by way of a ''Prime Element Theorem'' for distributive complete lattices with compact units.
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 03E25
zbMATH DE Number: 3920543
zbMATH Keywords: Almost Maximal Ideal Theorem
zbMATH Keywords: Prime Ideal Theorem
zbMATH Keywords: Axiom of Choice
zbMATH Keywords: Prime Element Theorem
zbMATH Keywords: distributive complete lattices with compact units
author: Bernhard Banaschewski
reviewed by: Peter Johnstone
MaRDI profile type: Publication
cites work: Lattice Aspects of Radical Ideals and Choice Principles
cites work: Prime ideals yield almost maximal ideals
cites work: Sur le théorème de Zorn
cites work: Q3965241
full work available at URL: https://doi.org/10.1007/bf00334858
OpenAlex ID: W2045952920
description: scientific article; zbMATH DE number 3920543