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| In [[abstract algebra]], an '''ordered ring''' is a [[commutative ring]] <math>R</math> with a [[total order]] ≤ such that for all ''a'', ''b'', and ''c'' in ''R'':
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| * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''.
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| * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''ab''.
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| Ordered rings are familiar from [[arithmetic]]. Examples include the [[real number]]s. (The rationals and reals in fact form [[ordered field]]s.) The [[complex number]]s do ''not'' form an ordered ring (or [[ordered field]]).
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| In analogy with real numbers, we call an element ''c'' ≠ 0, of an ordered ring '''positive''' if 0 ≤ ''c'' and '''[[negative number|negative]]''' if ''c'' ≤ 0. The set of positive (or, in some cases, nonnegative) elements in the ring ''R'' is often denoted by ''R''<sub>+</sub>.
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| If ''a'' is an element of an ordered ring ''R'', then the '''[[absolute value]]''' of ''a'', denoted |''a''|, is defined thus:
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| :<math>|a| := \begin{cases} a, & \mbox{if } 0 \leq a, \\ -a, & \mbox{otherwise}, \end{cases} </math>
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| where -''a'' is the [[additive inverse]] of ''a'' and 0 is the additive [[identity element]].
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| 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.
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| == Basic properties ==
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| if a < b and c > 0 then ac < bc is the correct definition.
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| For all ''a'', ''b'' and ''c'' in ''R'':
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| *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.
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| *|''ab''| = |''a''| |''b''|.<ref>OrdRing_ZF_2_L5</ref>
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| *An ordered ring that is not [[trivial ring|trivial]] is infinite.<ref>ord_ring_infinite</ref>
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| *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.
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| *An ordered ring ''R'' has no [[zero divisor]]s if and only if the positive ring elements are [[Closure (mathematics)|closed]] under multiplication (i.e. if ''a'' and ''b'' are positive, then so is ''ab'').<ref>OrdRing_ZF_3_L3</ref>
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| *In an ordered ring, no negative element is a square.<ref>OrdRing_ZF_1_L12</ref> This is because 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 positive.
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| ==Notes==
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| The names below refer to theorems formally verified by the [http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf IsarMathLib] project.
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| {{reflist}}
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| [[Category:Ordered groups]]
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| [[Category:Real algebraic geometry]]
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