\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.
@article{118738,
author = {Marcel Ern\'e},
title = {A primrose path from Krull to Zorn},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {36},
year = {1995},
pages = {123-126},
zbl = {0827.03028},
mrnumber = {1334420},
language = {en},
url = {http://dml.mathdoc.fr/item/118738}
}
Erné, Marcel. A primrose path from Krull to Zorn. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) pp. 123-126. http://gdmltest.u-ga.fr/item/118738/
A new proof that ``Krull implies Zorn'', preprint, McMaster University, Hamilton, 1993. | MR 1301940 | Zbl 0813.03032
On Krull's separation lemma, Order 10 (1993), 253-260. (1993) | MR 1267191 | Zbl 0795.06005
Krull implies Zorn, J. London Math. Soc. 19 (1979), 285-287. (1979) | MR 0533327 | Zbl 0394.03045
Commutative Rings, The University of Chicago Press, Chicago, 1974. | MR 0345945 | Zbl 0296.13001
Quantales and Their Applications, Pitman Research Notes in Mathematics Series 234, Longman Scientific and Technical, Essex, 1990. | MR 1088258 | Zbl 0703.06007
Equivalents of the Axiom of Choice, II, North-Holland, Amsterdam-New York-Oxford, 1985. | MR 0798475