Real Algebraic and Analytic Geometry
Submission: 2007, February 27.
A super real closed ring is a commutative ring which is equipped with the operation of all continuous functions Rn--> R. Examples are rings of continuous functions and super real fields attached to z-prime ideals in the sense of Dale and Woodin. We prove that super real closed rings which are fields are an elementary class of real closed fields, which carry all o-minimal expansions of the real field in a natural way.
The main part of the paper develops the commutative algebra of super real closed rings, by showing that many constructions of lattice ordered rings can be performed inside super real closed rings; the most important are: residue rings, complete and classical quotients, convex hulls, valuations, Prüfer hulls and real closures over proconstructible subsets.
We also give a counterexample to the conjecture that the first order theory of (pure) rings of continuous functions is the theory of real closed rings, which says in addition that a semi-local model is a product of fields .
Mathematics Subject Classification (2000): 13A10, 46E25, 54C05, 54C40.
Keywords and Phrases: Real closed rings, super real fields, rings of continuous functions, model theory, convexity, spectra.
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