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 $P (X)$ and the complete lattice of all conservative prime ideals of $R$. Moreover, the members of any system $S \subseteq P (X)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in $S=\bigcup \{RS:S\in S \}$, and the maximal members of $S $ correspond to the maximal ideals contained in $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.
Keywords: polynomial ring, conservative, prime ideal, system of finite character, Axiom of Choice
AMS Subject Classification: 03E25, 13B25, 13B30