Real closed * reduced partially ordered rings

TL;DR

This paper generalizes the concept of real closure * for commutative real rings, exploring maximal partial orderings, Baer rings, and essential extensions, and establishing bijections and topological characterizations of these structures.

Contribution

It introduces new characterizations and constructions of real closure * for real rings, including bijections with maximal partial orderings and topological insights.

Findings

Bijection between real closure * and maximal partial orderings.

Characterization of Baer reduced rings' integral closure.

Uniqueness of automorphisms of Baer hulls.

Abstract

In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also make some very interesting and significant discoveries regarding maximal partial orderings of rings, Baer rings and essentail extension of rings. The first Theorem itself gives us a noteworthy bijection between maximal partial orderings of two rings by which one is a rational extension of the other. We characterize conditions when a Baer reduced ring can be integrally closed in its total quotient ring. We prove that Baer hulls of rings have exactly one automorphism (the identity) and we even prove this for a general case (Lemma 12). Proposition 14 allows us to study essential extensions of rings and their relation with minimal prime spectrum of the lower ring. And Theorem 15 gives us a construction of the real spectrum of a ring generated by adjoining idempotents to a reduced commutative subring (for instance the construction of Baer hull of reduced commutative rings). From most of the above interesting theories we prove that there is a bijection between the real closure * of real rings (upto isomorphisms) and their maximal partial orderings. We then attempt to develop some topological theories for the set of real closure * of real rings (upto isomorphism) that will help us give a topological characterization in terms of the real and prime spectra of these rings. The topological characterization will be revealed in a later work. It is noteworthy to point out that we can allow ourself to consider mostly the minimal prime spectrum of the real ring in order to develop our topological theories.