abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, an ordered ring is a (usually

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

) ring ''R'' with a total order ≤ 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 branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...

. Examples include the

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s, the rationals 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s. (The rationals and reals in fact form ordered fields.) The

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, 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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

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 an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero el ...

a

and 0 is the additive

identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

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 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 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'' = ''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