Ordered ring

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Ordered ring

In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, an ordered ring is a commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a Ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
with a total order

Total order

In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X....
= such that for all a, b, and c in R:

Ordered rings are familiar from arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations....
. Examples include the real number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s. (The rationals and reals in fact form ordered field

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations....
s.) The complex number

Complex number

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In abstract algebra

Abstract algebra

Commutative ring

Total order

In mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some Set X....
= 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.

Ordered rings are familiar from arithmetic

Arithmetic

Real number

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations....
s.) The complex number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s do not form an ordered ring (or ordered field).

In analogy with real numbers, we call an element c ? 0, of an ordered ring positive if 0 = c and negative
Negative and non-negative numbers

A negative number is a real number that is inequality 0 , such as -3. A positive number is a real number that is greater than zero, such as 2....
if c = 0. The set of positive (or, in some cases, nonnegative) elements in the ring R is often denoted by R₊.

If a is an element of an ordered ring R, then the absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
of a, denoted |a|, is defined thus:

where -a is the additive inverse

Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
of a and 0 is the additive identity element

Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
.

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
Trivial ring

In mathematics, a trivial ring is a ring defined on a singleton set, . The ring operations are trivial:One often refers to the trivial ring since every trivial ring is Ring isomorphism to any other ....
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
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
, linearly ordered group
Linearly ordered group

In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the partially ordered set "≤" is total order....
s with respect to addition.
An ordered ring R has no zero divisor
Zero divisor

In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Right zero divisors are defined analogously, that is, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0....
s if and only if the positive ring elements are closed
Closure (mathematics)

In mathematics, a Set is said to be closed under some operation if the Operation on members of the set produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not....
under multiplication (i.e. if a and b are positive, then so is ab).
In an ordered ring, no negative element is a square. This is because if a ? 0 and a = b² then b ? 0 and a = (-b )²; as either b or -b is positive, a must be positive.