Super real closed rings

Marcus Tressl¹

¹ Universität Passau, IM Innstr. 33 D-94032 Passau, Germany

Fundamenta Mathematicae, Tome 194 (2007) no. 2, pp. 121-177

Résumé

A super real closed ring is a commutative ring equipped with the operation of all continuous functions ${\mathbb R}^n\to {\mathbb R}$. Examples are rings of continuous functions and super real fields attached to $z$-prime ideals in the sense of Dales 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.

Détail

DOI : 10.4064/fm194-2-2

Keywords: super real closed ring commutative ring equipped operation continuous functions mathbb mathbb examples rings continuous functions super real fields attached z prime ideals sense dales woodin prove super real closed rings which fields elementary class real closed fields which carry o minimal expansions real field natural main part paper develops commutative algebra super real closed rings showing many constructions lattice ordered rings performed inside super real closed rings important residue rings complete classical quotients convex hulls valuations fer hulls real closures proconstructible subsets counterexample conjecture first order theory pure rings continuous functions theory real closed rings which says addition semi local model product fields