Summary: 1. The real numbers.
1.1. Ordered rings.
Definition 1.1. By an ordered commutative ring with unity we mean an
(R, +, 0, , 1, P)
such that (R, +, 0, , 1) is a commutative ring with unity and such that P is a subset
of R with the following properties:
(i) if x R the exactly one of the following holds:
-x P; x = 0; x P;
(ii) if x P and y P then x + y P;
(iii) if x P and y P then xy P.
If x P we say x is positive and if -x P we say x is negative.
We will frequently say "R is an ordered ring with positive elements P" instead
of "(R, , 0, µ, 1, P) is an ordered ring".
Suppose R is an ordered ring with positive elements P.
Proposition 1.1. Suppose x, y R and xy = 0. Then either x = 0 or y = 0.
Remark 1.1. That is, R is an integral domain.
Proof. We have
x P and y P xy P xy = 0;
x P and - y P -(xy) = x(-y) P xy = 0;