A Stone ring is a partially
ordered ring K with unil element 1 satisfying (1) 1 is positive; (2) for every x in K
there exists a natural number n such that n⋅ 1 −x belongs to K; and (3) if 1 + nx is
positive for all natural numbers n then x is positive. Our first theorem: Every Stone
ring is order-isomorphic with a subring of the ring of all continuous real functions on
some compact Hausdorff space, with the usual partial order. A corollary
is a theorem first proved by Harrison: Let K be a partially ordered ring
satisfying conditions (1) and (2), and suppose the positive cone of K is maximal
in the family of all subsets of K which exclude −1 and are closed under
addition and multiplication. Then K is order-isomorphic with a subring of the
reals.
