Editing Ordered ring

[[Image:Real number line.svg|thumb|right|350px|The [[real number]]s 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|commutative]]) [[ring (mathematics)|ring]] ''R'' with a [[total order]] ≤ such that for all ''a'', ''b'', and ''c'' in ''R'':<ref>{{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 }}</ref>

* 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 [[integer]]s, the [[rational number|rational]]s and the [[real number]]s.<ref>*{{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 }}
</ref>  (The rationals and reals in fact form [[ordered field]]s.) The [[complex number]]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''<sub>+</sub>. An alternative notation, favored in some disciplines, is to use ''R''<sub>+</sub> for the set of nonnegative elements, and ''R''<sub>++</sub> for the set of positive elements.

==Absolute value==

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 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''.<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''|{{Hair space}}|''b''|.<ref>OrdRing_ZF_2_L5</ref>
*An ordered ring that is not [[trivial ring|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 group|abelian]], [[linearly ordered group]]s 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'' = ''b''<sup>2</sup> then ''b'' ≠ 0  and ''a''  = (−''b'')<sup>2</sup>; as either ''b'' or −''b'' is positive, ''a'' must be nonnegative.

== See also ==

* {{annotated link|Ordered field}}
* {{annotated link|Ordered group}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Ordered vector space}}
* {{annotated link|Partially ordered ring}}
* {{annotated link|Partially ordered space}}
* {{annotated link|Riesz space}}, also called vector lattice
* [[Semiring#Ordered_semirings|Ordered semirings]]

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

[[Category:Ordered groups]]
[[Category:Real algebraic geometry]]