Orderings for noncommutative rings

and Notation. The goal of this paper is to present the beginnings of a theory of real algebraic geometry for noncommutative rings. For a basic introduction to the commutative theory, see Lam [L]. The word field will be used in this paper to mean a (generally noncommutative) skewfield; we shall specify a commutative field when we need to. R will denote a noncommutative ring with 1. We shall define a concept of ordering for R which we show behaves properly with respect to orderings of “residue fields” and generalizes the usual concepts of orderings for fields and commutative rings. In the final section, we take a brief look at the real spectrum. The complications of this theory can be avoided for special classes of noncommutative rings such as Ore domains (see, for example, [P]). Let M(R) denote the set of all square matrices over R. The notation (a|A) will be used to denote an augmented matrix with a as the first column and A as the remainder of the matrix. The basic concepts for noncommutative algebraic geometry which we use are due to P. M. Cohn [Co1, Co2, Co3] and we shall generally use his notation. In addition to the usual matrix operations, we will use