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Keywords:
polynomial ring; conservative; prime ideal; system of finite character; Axiom of Choice
polynomial ring; conservative; prime ideal; system of finite character; Axiom of Choice
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.
References:
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