Positivity 11 (2007), 639–686
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040639-48, published online September 26, 2007
Positivity and Convexity in Rings of Fractions
Abstract. Given a commutative ring A equipped with a preordering A
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
extends to a preordering R
of R (i.e. with A ∩ R
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
Mathematics Subject Classiﬁcation (2000). Primary 13J25, 13J30; Secondary
Keywords. Preordered ring extension, positively dense set, convexity cover,
positivity divisor, convexity divisor.
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
, but −1 ∈ A
. In other words,
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 x
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
. 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 rea-
sons 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.
Supported by DFG.
A short form of this article has been delivered at the conference Carthapos 2006 at Carthago