Ordered ring


In abstract algebra, an ordered ring is a ring R with a total order ≤ such that for all a, b, and c in R:
Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers. The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and i.

Positive elements

In analogy with the real numbers, we call an element c of an ordered ring R positive if 0 < c, and negative if c < 0. 0 is considered to be neither positive nor negative.
The set of positive elements of an ordered ring R is often denoted by R+. An alternative notation, favored in some disciplines, is to use R+ for the set of nonnegative elements, and R++ for the set of positive elements.

Absolute value

If is an element of an ordered ring R, then the absolute value of, denoted, is defined thus:
where is the additive inverse of and 0 is the additive identity element.

Discrete ordered rings

A discrete ordered ring or discretely ordered ring is an ordered ring in which there is no element between 0 and 1. The integers are a discrete ordered ring, but the rational numbers are not.

Basic properties

For all a, b and c in R: