  We examine several classical concepts from topology and functional analysis,
using methods of commutative algebra. We show that these various concepts are
all controlled by BC R-rings and their maximal spectra.
  A BC R-ring is a ring A that is isomorphic to the ring of bounded continuous
R-valued functions on some compact topological space X. These rings are not
topologized.
  We prove that the category of BC R-rings is dual to the category of compact
topological spaces. Next we prove that for every topological space X the ring
of bounded continuous functions on it is a BC R-ring. These theorems combined
yield an algebraic construction of the Stone-Cech Compactification of an
arbitrary topological space.
  There is a similar notion of BC C-ring. Every BC C-ring A has a canonical
involution. The canonical hermitian subring of A is a BC R-ring, and this is an
equivalence of categories from BC C-rings to BC R-rings.
  Let K be either R or C. We prove that a BC K-ring A has a canonical norm on
it, making it into a Banach K-ring. We then prove that the forgetful functor is
an equivalence from Banach^* K-rings (better known as commutative unital C^*
K-algebras) to BC K-rings. The quasi-inverse of the forgetful functor endows a
BC K-ring with its canonical norm, and the canonical involution when K = C.
  Stone topological spaces, also known as profinite topological spaces, are
traditionally related to boolean rings - this is Stone Duality. We give a BC
ring characterization of Stone spaces. From that we obtain a very easy proof of
the fact that the Stone-Cech Compactification of a discrete space is a Stone
space.
  Most of the results in this paper are not new. However, most of our proofs
seem to be new - and our methods could potentially lead to genuine progress
related to these classical topics.
