positive cone

Let R be a commutative ring with 1. A subset P of R is called a pre-positive cone of R provided that

  1. 1.

    P+PP (P is additively closed)

  2. 2.

    PPP (P is multiplicatively closed)

  3. 3.


  4. 4.


As it turns out, a field endowed with a pre-positive cone has an order structureMathworldPlanetmath. The field is called a formally real (http://planetmath.org/FormallyRealField), orderable, or ordered field. Before defining what this “order” is, let’s do some preliminary work. Let P0 be a pre-positive cone of a field F. By Zorn’s Lemma, the set of pre-positive cones extending P0 has a maximal elementMathworldPlanetmath P. It can be shown that P has two additional properties:

  1. 5.


  2. 6.



First, suppose there is aF-(P(-P)). Let P¯=P+Pa. Then aP¯ and so P is strictly contained in P¯. Clearly, sqr(F)P¯ and P¯ is easily seen to be additively closed. Also, P¯ is multiplicatively closed as the equation (p1+q1a)(p2+q2a)=(p1p2+q1q2a2)+(p1q2+q1p2)a demonstrates. Since P is a maximal and P¯ properly contains P, P¯ is not a pre-positive cone, which means -1P¯. Write -1=p+qa. Then q(-a)=p+1P. Since qP, 1/q=q(1/q)2P, -a=(1/q)(p+1)P, contradicting the assumptionPlanetmathPlanetmath that a-P. Therefore, P(-P)=F.

For the second part, suppose aP(-P). Since a-P, -aP. If a0, then -1=a(-a)(1/a)2P, a contradictionMathworldPlanetmathPlanetmath. ∎

A subset P of a field F satisfying conditions 1, 2, 5 and 6 is called a positive conePlanetmathPlanetmathPlanetmath of F. A positive cone is a pre-positive cone. If aF, then either aP or -aP. In either case, a2P. Next, if -1P, then 1-P. But 1=12P, we have 1P(-P), contradicting Condition 6 of P.

Now, define a binary relationMathworldPlanetmath , on F by:


It is not hard to see that is a total orderMathworldPlanetmath on F. In additionPlanetmathPlanetmath, with the additive and multiplicative structures on F, we also have the following two rules:

  1. 1.


  2. 2.

    0a and 0b0ab.

Thus, F is a field ordered by .

Remark. Positive cones may be defined for more general ordered algebraic structuresPlanetmathPlanetmath, such as partially ordered groups, or partially ordered rings.


  • 1 A. Prestel, Lectures on Formally Real Fields, Springer, 1984
Title positive cone
Canonical name PositiveCone
Date of creation 2013-03-22 14:46:54
Last modified on 2013-03-22 14:46:54
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 13J25
Classification msc 12D15
Related topic PositivityInOrderedRing
Related topic FormallyRealField
Defines pre-positive cone