Orderings and maximal ideals of rings of analytic functions

Abstract

We prove that there is a natural injective correspondence between the maximal ideals of the ring of analytic functions on a real analytic set X X and those of its subring of bounded analytic functions. By describing the maximal ideals in terms of ultrafilters we see that this correspondence is surjective if and only if X X is compact. This approach is also useful for studying the orderings of the field of meromorphic functions on X X .