  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.
