A note on David Harrison’s theory of preprimes

A Stone ring is a partially ordered ring K with unit element 1 satisfying (1) 1 is positive; (2) for every x in K there exists a natural number n such that nx 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.