Ordered ring

File:Real number line.svg

The real numbers are an ordered ring which is also an ordered field. The integers, a subset of the real numbers, are an ordered ring that is not an ordered field.

In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R:<ref>Template:Citation</ref>

if a ≤ b then a + c ≤ b + c.
if 0 ≤ a and 0 ≤ b then 0 ≤ ab.

ExamplesEdit

Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers.<ref>*Template:Citation </ref> (The rationals and reals in fact form ordered fields.) 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 elementsEdit

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 valueEdit

If <math> a</math> is an element of an ordered ring R, then the absolute value of <math>a</math>, denoted <math>|a|</math>, is defined thus:

<math>|a| := \begin{cases} a, & \mbox{if } 0 \leq a, \\ -a, & \mbox{otherwise}, \end{cases} </math>

where <math>-a</math> is the additive inverse of <math>a</math> and 0 is the additive identity element.

Discrete ordered ringsEdit

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 propertiesEdit

For all a, b and c in R:

If a ≤ b and 0 ≤ c, then ac ≤ bc.<ref>OrdRing_ZF_1_L9</ref> This property is sometimes used to define ordered rings instead of the second property in the definition above.
|ab| = |a|Template:Hair space|b|.<ref>OrdRing_ZF_2_L5</ref>
An ordered ring that is not trivial is infinite.<ref>ord_ring_infinite</ref>
Exactly one of the following is true: a is positive, −a is positive, or a = 0.<ref>OrdRing_ZF_3_L2, see also OrdGroup_decomp</ref> This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
In an ordered ring, no negative element is a square:<ref>OrdRing_ZF_1_L12</ref> Firstly, 0 is square. Now if a ≠ 0 and a = b² then b ≠ 0 and a = (−b)²; as either b or −b is positive, a must be nonnegative.

NotesEdit

The list below includes references to theorems formally verified by the IsarMathLib project. Template:Reflist