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Title: A primrose path from Krull to Zorn (English)
Author: Erné, Marcel
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 36
Issue: 1
Year: 1995
Pages: 123-126
Category: math
Summary: \font\jeden=rsfs10 \font\dva=rsfs8 \font\tri=rsfs6 \font\ctyri=rsfs7 Given a set $X$ of ``indeterminates'' and a field $F$, an ideal in the polynomial ring $R=F[X]$ is called conservative if it contains with any polynomial all of its monomials. The map $S\mapsto RS$ yields an isomorphism between the power set $\text{\dva P}\,(X)$ and the complete lattice of all conservative prime ideals of $R$. Moreover, the members of any system $\text{\dva S}\,\subseteq \text{\dva P}\,(X)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in $P_{\text{\ctyri S}}=\bigcup \{RS:S\in \text{\dva S}\,\}$, and the maximal members of $\text{\dva S}\,$ correspond to the maximal ideals contained in $P_{\text{\ctyri S}}\,$. This establishes, in a straightforward way, a ``local version'' of the known fact that the Axiom of Choice is equivalent to the existence of maximal ideals in non-trivial (unique factorization) rings. (English)
Keyword: polynomial ring
Keyword: conservative
Keyword: prime ideal
Keyword: system of finite character
Keyword: Axiom of Choice
MSC: 03E25
MSC: 04A25
MSC: 13A15
MSC: 13B25
MSC: 13B30
MSC: 13F20
idZBL: Zbl 0827.03028
idMR: MR1334420
Date available: 2009-01-08T18:16:31Z
Last updated: 2012-04-30
Stable URL:
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Reference: [2] Banaschewski B., Erné M.: On Krull's separation lemma.Order 10 (1993), 253-260. Zbl 0795.06005, MR 1267191
Reference: [3] Hodges W.: Krull implies Zorn.J. London Math. Soc. 19 (1979), 285-287. Zbl 0394.03045, MR 0533327
Reference: [4] Kaplansky I.: Commutative Rings.The University of Chicago Press, Chicago, 1974. Zbl 0296.13001, MR 0345945
Reference: [5] Rosenthal K.: Quantales and Their Applications.Pitman Research Notes in Mathematics Series 234, Longman Scientific and Technical, Essex, 1990. Zbl 0703.06007, MR 1088258
Reference: [6] Rubin H., Rubin J.E.: Equivalents of the Axiom of Choice, II.North-Holland, Amsterdam-New York-Oxford, 1985. MR 0798475


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