Totally t-closed rings

Real Closed Rings and Real Closed * Rings

Here we try to distinguish and compare different notions of real closedness mainly one developed by N. Schwartz in his Habilitationschrift and the other developed by A. Sankaran and K. Varadarajan in [SV] which we shall call real closed *. We stick to the definition of real closed rings as defined and characterized in [RCR] and we try to determine and characterize real closed rings that are rea...

Real closed valuation rings

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of re...

Super real closed rings

Contents 1. Introduction 2. Real closed rings-a model theoretic tour 3. Computation of the z-radical in C(X) 4. A sentence in the language of rings separating continuous semi-algebraic from arbitrary continuous functions 5. Super real closed rings: Definition and basic properties 6. Υ-ideals 7. Localization of super real closed rings 8. Application: o-minimal structures on super real closed fie...

Bounded super real closed rings

Division closed partially ordered rings

Fuchs [6] called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and ...