Super real closed rings

Abstract

Abstract. A super real closed ring is a commutative ring equipped with the operation of all continuous functions R n → 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.

Bounded super real closed rings

“…The proof is parallel to the proof of the localization theorem [Tr2]:(7.4), using (3.5) instead of [Tr2]:(7.2)(i). Proof.…”

“…However, any other super real closed ring B expanding the pure ring having A as a bounded super real closed subring, has Hol A as a super real closed subring (cf. [Tr2]:(9.2)(i)) and the underlying bounded super real closed ring structure is the one induced from A. By (3.4), the super real closed ring structures of B and induced on Hol A are equal.…”

“…Existence is given by [Tr2]:(7.2)(ii). Uniqueness holds, since G(x 1 , ..., x n , y) is uniquely determined by ( * ) for all (x, y) ∈ IR n × IR with y = 0.…”

“…This note is a complement to the paper [Tr2], where super real closed rings are introduced and studied. A super real closed ring A is a commutative unital ring A together with an operation F A : A n −→ A for every continuous map F : IR n −→ IR, n ∈ IN, so that all term equalities between the F 's remain valid for the F A 's.…”

“…The results mentioned above can be used to transfer most of the commutative algebra, developed in [Tr2] sections 11, 12 and 13 (with the appropriate adaptions) to bounded super real closed rings. It would be tedious to elaborate this here, instead we present instruments which allow such a transfer easily, whenever it is needed in subsequent work.…”