Real Algebraic and Analytic Geometry |

Real closed * reduced partially ordered Rings.

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Submission: 2009, June 1.

*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.*

Mathematics Subject Classification (2000): 13J25, 06E15, 16E50.

Keywords and Phrases: real closed * rings, regular rings, absolutes of Hausdorff spaces, irreducible surjections, f-ring partial orderings,total quotient ring, maximal partial ordering of rings, essential extensions of rings.

**Full text**, 17p.:
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