| 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:
|
http://hdl.handle.net/10338.dmlcz/118738 |
| . |
| Reference:
|
[1] Banaschewski B.: A new proof that ``Krull implies Zorn''.preprint, McMaster University, Hamilton, 1993. Zbl 0813.03032, MR 1301940 |
| 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 |
| . |
