abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...

, an ordered ring is a (usually commutative) ring ''R'' with a

total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...

≤ such that for all ''a'', ''b'', and ''c'' in ''R'': * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''. * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''ab''.

Examples

Ordered rings are familiar from

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...

. Examples include the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

s, the

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...

s and the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...

s. (The rationals and reals in fact form

ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...

s.) 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

a

is an element of an ordered ring ''R'', then the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

a

, denoted

, a,

, is defined thus: :

, a,  := \begin a, & \mbox  0 \leq a,  \\ -a,  & \mbox, \end

where

-a

is the

additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...

a

and 0 is the additive

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...

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''. This property is sometimes used to define ordered rings instead of the second property in the definition above. *, ''ab'', = , ''a'', , ''b'', . *An ordered ring that is not

trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

is infinite. *Exactly one of the following is true: ''a'' is positive, -''a'' is positive, or ''a'' = 0. This property follows from the fact that ordered rings are abelian,

linearly ordered group In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a: * le ...

s with respect to addition. *In an ordered ring, no negative element is a square.OrdRing_ZF_1_L12 This is because if ''a'' ≠ 0 and ''a'' = ''b''² 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 th
IsarMathLib
project. {{reflist Ordered groups Real algebraic geometry