ordered ring

Image:Real number line.svgs 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:{{citation | last=Lam | first=T. Y. | authorlink=Tsit Yuen Lam | title=Orderings, valuations and quadratic forms | series=CBMS Regional Conference Series in Mathematics | volume=52 | publisher=American Mathematical Society | year=1983 | isbn=0-8218-0702-1 | zbl=0516.12001 | url-access=registration | url=https://archive.org/details/orderingsvaluati0000lamt }}

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

Examples

Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers.*{{citation|author=Lam, T. Y.|authorlink=Tsit Yuen Lam |title=A first course in noncommutative rings |series=Graduate Texts in Mathematics|volume=131 |edition=2nd |publisher=Springer-Verlag |place=New York |year=2001 |pages=xx+385 |isbn=0-387-95183-0 |mr=1838439 | zbl=0980.16001 }}

(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 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 $a$ is an element of an ordered ring R, then the absolute value of $a$ , denoted $|a|$ , is defined thus:

: $|a| := \begin{cases} a, & \mbox{if } 0 \leq a, \\ -a, & \mbox{otherwise}, \end{cases}$

where $-a$ is the additive inverse of $a$ 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:

If a ≤ b and 0 ≤ c, then ac ≤ bc.OrdRing_ZF_1_L9 This property is sometimes used to define ordered rings instead of the second property in the definition above.
|ab| = |a|{{Hair space}}|b|.OrdRing_ZF_2_L5
An ordered ring that is not trivial is infinite.ord_ring_infinite
Exactly one of the following is true: a is positive, −a is positive, or a = 0.OrdRing_ZF_3_L2, see also OrdGroup_decomp 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:OrdRing_ZF_1_L12 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.

Notes

The list below includes references to theorems formally verified by the [http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf IsarMathLib] project.

Category:Ordered groups

Category:Real algebraic geometry