# A primrose path from Krull to Zorn

Commentationes Mathematicae Universitatis Carolinae (1995)

- Volume: 36, Issue: 1, page 123-126
- ISSN: 0010-2628

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topErné, 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 -

## References

top- Banaschewski B., A new proof that ``Krull implies Zorn'', preprint, McMaster University, Hamilton, 1993. Zbl0813.03032MR1301940
- Banaschewski B., Erné M., On Krull's separation lemma, Order 10 (1993), 253-260. (1993) Zbl0795.06005MR1267191
- Hodges W., Krull implies Zorn, J. London Math. Soc. 19 (1979), 285-287. (1979) Zbl0394.03045MR0533327
- Kaplansky I., Commutative Rings, The University of Chicago Press, Chicago, 1974. Zbl0296.13001MR0345945
- Rosenthal K., Quantales and Their Applications, Pitman Research Notes in Mathematics Series 234, Longman Scientific and Technical, Essex, 1990. Zbl0703.06007MR1088258
- Rubin H., Rubin J.E., Equivalents of the Axiom of Choice, II, North-Holland, Amsterdam-New York-Oxford, 1985. MR0798475

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