Orderings, Real Places and Valuations on Noncommutative Integral Domains

This paper is a continuation of our recent joint work with K.H. Leung on the real spectrum of a noncommutative ring 15]. The object of the present paper is to develop a theory of real places and valuations to accompany the theory of orderings developed in 15]. Because of the existence of integral domains which are not embeddable in a skew eld, it is necessary to deal directly with integral domains. Consequently, the theory of real places and valuations that we obtain is somewhat less precise than the corresponding theory for elds or skew elds. Archimedian classes in ordered abelian groups were considered already by Hahn 11] and the connection between orderings and real places on elds was worked out already by Baer and Krull in 2] and 12]. In the 1970s, inspired by PPster's earlier work on signatures of quadratic forms 17], the connection between orderings, valuations and quadratic forms on elds was rmly established by Becker, Brr ocker, Brown, Prestel and others; see 13]. In the 1980s, after the real spectrum of a ring was introduced by Coste and Roy, these ideas were applied to real algebraic geometry and real analytic geometry; see 3] and 1]. Abstract real spectra, also called spaces of signs, were introduced just recently in 1] 5] and 16] in an attempt to axiomatize parts of real algebraic geometry and real analytic geometry. In 15] it is shown, and perhaps this is a bit surprising (although there is some hint of it already in 14, Chapter 6] and in 8], for example), that the orderings on a noncommutative ring form an abstract real spectrum, exactly as in the commutative case. In the course of the proof, it is shown that if p is a real prime of A, then A=p is an integral domain and the orderings on A having support p form a space of orderings. In the present paper, we deene real places on a (not necessarily commutative) integral domain A and examine the relationship between real places on A and support f0g orderings on A. We show that a version of the Baer-Krull theorem holds, and that the real places yield a natural P-structure on the space of support f0g orderings. (See 16] for the meaning of this terminology.) More generally, we show, for an arbitrary noncommutative ring, that the real places on the various residue domains yield a natural P-structure …