Uniqueness of real closure * of regular rings

TL;DR

This paper characterizes the real closure * of regular rings, compares it with Baer regular rings, and classifies their spectra using topological and continuous section methods, revealing new structural insights.

Contribution

It provides a new characterization of real closure * for regular rings and introduces methods to classify spectra of Baer and non-Baer regular rings.

Findings

Characterization of real closure * of regular rings.

Classification of spectra using continuous sections.

Example of a Baer regular ring not rationally complete.

Abstract

In this paper we give a characterisation of real closure * of regular rings, which is quite similar to the characterisation of real closure * of Baer regular rings seen in [4]. We also characterize Baer-ness of regular rings using near-open maps. The last part of this work will concentrate on classifying the real closure * of Baer and non-Baer regular rings (upto isomorphisms) using continuous sections of the support map, we construct a topology on this set for the Baer case. For the case of non-Baer regular rings, it will be shown that almost no information of the ring structure of the Baer hull is necessary in order to study the real and prime spectra of the Baer hull. We shall make use of the absolutes of Hausdorff spaces in order to give a construction of the spectra of the Baer hulls of regular rings. Finally we give example of a Baer regular ring that is not rationally complete.