The real spectrum of a noncommutative ring
Orderings on commutative rings have been studied extensively in the context of real algebraic geometry and, in particular, the study of semi-algebraic sets, since the early work by Coste and Roy (in the early 80s) on the real spectrum. In particular, it is now well-known that orderings on a real commutative ring yield an abstract real spectrum (or space of signs), in the sense of Bröckner et al. This is basically due to the fact that, on the one hand, the axioms defining real spectra generalize those for spaces of orderings, while on the other the residue space \((X_p,G_p)\) at any prime \(p\) of an abstract real spectrum \((X,G)\) is a space of orderings. The main purpose of the present paper is to take a look at what happens in the noncommutative case. The surprising, main result in the present paper is that it may be shown that orderings of noncommutative rings give rise to abstract real spectra, just as in the commutative case.
scientific article; zbMATH DE number 1106617
scientific article