A primrose path from Krull to Zorn

Erné, Marcel. "A primrose path from Krull to Zorn." Commentationes Mathematicae Universitatis Carolinae 36.1 (1995): 123-126. <http://eudml.org/doc/247750>.

@article{Erné1995,
abstract = {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 $\text\{P\}\,(X)$ and the complete lattice of all conservative prime ideals of $R$. Moreover, the members of any system $\text\{S\}\,\subseteq \text\{P\}\,(X)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in $P_\{\text\{S\}\}=\bigcup \lbrace RS:S\in \text\{S\}\,\rbrace $, and the maximal members of $\text\{S\}\,$ correspond to the maximal ideals contained in $P_\{\text\{S\}\}\,$. 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.},
author = {Erné, Marcel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {polynomial ring; conservative; prime ideal; system of finite character; Axiom of Choice; conservative prime ideal; pseudo-localization by a primrose; Krull's maximal ideal theorem for unique factorization domains; axiom of choice},
language = {eng},
number = {1},
pages = {123-126},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A primrose path from Krull to Zorn},
url = {http://eudml.org/doc/247750},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Erné, Marcel
TI - A primrose path from Krull to Zorn
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 1
SP - 123
EP - 126
AB - 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 $\text{P}\,(X)$ and the complete lattice of all conservative prime ideals of $R$. Moreover, the members of any system $\text{S}\,\subseteq \text{P}\,(X)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in $P_{\text{S}}=\bigcup \lbrace RS:S\in \text{S}\,\rbrace $, and the maximal members of $\text{S}\,$ correspond to the maximal ideals contained in $P_{\text{S}}\,$. 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.
LA - eng
KW - polynomial ring; conservative; prime ideal; system of finite character; Axiom of Choice; conservative prime ideal; pseudo-localization by a primrose; Krull's maximal ideal theorem for unique factorization domains; axiom of choice
UR - http://eudml.org/doc/247750
ER -