Real Algebraic and Analytic Geometry
Submission: 2002, December 18.
We prove that there is a natural injective correspondence between the maximal ideals of the ring of analytic functions on a real analytic set $X$ 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 $X$ is compact. This approach is also useful to study the orderings of the field of meromorphic functions on $X$.
Mathematics Subject Classification (2000): 14P15, 32B15, 32B20.
Keywords and Phrases: Real analytic sets, analytic functions, maximal ideals, ultrafilters, orderings.
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