The Real Spectrum of a Noncommutative Ring

Abstract

Spaces of orderings were introduced by the second author in a series of w x papers 15]19 and various structure results were obtained generalizing results proved earlier for formally real fields by various people: E. Becker, w x L. Brocker, R. Brown, T. Craven, A. Prestel, and N. Schwartz. In 6, 26 Ž w x. also see 20, 23, 25 Craven and Tschimmel prove that formally real skew w x Ž w x. fields give rise to spaces of orderings. In 8 also see 9 Kalhoff extends Ž . w x Ž this to ternary fields also called planar ternary rings . In 10, 11 also see w x. 28 Kleinstein and Rosenberg and Knebusch, respectively, show how Ž . spaces of orderings arise in the study of real commutative semi-local rings. Orderings on commutative rings have been studied extensively since the introduction of the real spectrum by Coste and Roy in the early 1980s.