The Real Spectrum of a Noncommutative Ring

Spaces of orderings were introduced by the second author in a series of papers 15]{{19] and various structure results were obtained generalizing results proved earlier for formally real elds by various people: In 6] and 26] (also see 20] and 23] 25]) Craven and Tschimmel prove that formally real skew elds give rise to spaces of orderings. In 8] (also see 9]) Kalhoo extends this to ternary elds (also called planar ternary rings). In 10] and 11] (also see 28]) Kleinstein and Rosenberg and Knebusch respectively show how 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 1980's. Here, the main motivation comes from real algebraic geometry and the study of semialgebraic sets; see 2] 12]. Abstract real spectra (also called spaces of signs) were introduced just recently in 1] 4] and 21]. Orderings on a real commutative ring give rise to an abstract real spectrum. The axioms of an abstract real spectrum generalize those of a space of orderings. Conversely, if (X; G) is an abstract real spectrum then G has prime ideals and at each prime p G we can form the so-called residue space (X p ; G p) which is a space of orderings. In 1] 21] various local-global principles are proved for abstract real spectra. In particular, the results on minimal generation of semialgebraic sets due to L. Brr ocker and C. Scheiderer carry over to this abstract setting. Ordered skew elds were considered already by D. Hilbert in connection with his work on the foundations of geometry. Orderings on general noncommutative rings have received less attention. In 13, Chapter 6], Lam proves basic properties and gives some history; also see 7]. In 24], Powers introduces the real spectrum of higher level of a noncommutative ring. In the present paper we show that orderings (of level 1) on noncommutative rings give rise to abstract real spectra exactly as in the commutative case. As part of the proof, we show that if p is a real prime in a noncommutative ring A, then the orderings on A having support p form a space of orderings in a natural way. We show that if p is a real prime of A, then ab 2 p) a 2 p or b 2 p …