A primrose path from Krull to Zorn

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 7→ RS yields an isomorphism between the power setP (X) and the complete lattice of all conservative prime ideals of R. Moreover, the members of any system S ⊆ P (X) of finite character are in one-to-one correspondence with the conservative prime ideals contained in PS = S {RS : S ∈ S }, and the maximal members of S correspond to the maximal ideals contained in PS . 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.