Positivity and convexity in rings of fractions∗)

Given a commutative ring A equipped with a preordering A (in the most general sense, see below), we look for a fractional ring extension (= “ring of quotients” in the sense of Lambek et al. [L]) as big as possible such that A extends to a preordering R of R (i.e. with A ∩R+ = A) in a natural way. We then ask for subextensions A ⊂ B of A ⊂ R such that A is convex in B with respect to B := B ∩R+. Introduction; partially ordered integral domains In this paper a “ring” always means a commutative ring with 1. Often a ring A will be preordered, i.e. there is given a set A+ ⊂ A with x + y ∈ A+, xy ∈ A+ for any two elements x, y of A+, 0 ∈ A+, 1 ∈ A+, but −1 6∈ A+. In other words, A+ is a semiring in A not containing −1. One should think of A+ as the set of “non-negative” elements of A. This is a very general notion of preordering. In the literature most often it is assumed that x2 ∈ A+ for every x ∈ A. Such a preordering will be called quadratic. More generally A+ will be called torsion, if for every x ∈ A there exists an even power x2n ∈ A+. Torsion preorderings play an important role in studies on sums of even powers and related topics, cf. e.g. [Be], [BeG], [Ber], [BW]. But for reasons which will become clear later, we will be compelled to admit also non-torsion preorderings; and anyway, there exist preorderings very relevant in real algebra, which are non-torsion, see below. If f, g are elements of a preordered ring A = (A,A+) then we decree that f ≤ g iff g − f ∈ A+. This is a (partial) ordering of the set A in the common sense only if A+ ∩ (−A+) = {0}, and consequently we then call A+ itself a partial ordering of A. Nowadays it is plain in real algebra that one has to study also preorderings if one is only interested in partial orderings. Given a preordered ring A = (A,A+) we will pursue two goals: Firstly we look for a ring R ⊃ A within the complete ring of quotients Q(A) of A (cf. §2 below) as big as possible, such that A+ ∗) Supported by DFG. A short form of this article has been delivered at the conference Carthapos 2006 at Carthago (Tunisia). 2000 Mathematics Subject Classification: Primary 13J25, 13J30; Secondary 13F05.