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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'': | |||
* if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''. | |||
* if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''ab''. | |||
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]]). | |||
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>. | |||
If ''a'' is an element of an ordered ring ''R'', then the '''[[absolute value]]''' of ''a'', denoted |''a''|, is defined thus: | |||
:<math>|a| := \begin{cases} a, & \mbox{if } 0 \leq a, \\ -a, & \mbox{otherwise}, \end{cases} </math> | |||
where -''a'' is the [[additive inverse]] of ''a'' and 0 is the additive [[identity element]]. | |||
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 == | |||
if a < b and c > 0 then ac < bc is the correct definition. | |||
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''| |''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. | |||
*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> | |||
*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. | |||
==Notes== | |||
The names below refer to theorems formally verified by the [http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf IsarMathLib] project. | |||
{{reflist}} | |||
[[Category:Ordered groups]] | |||
[[Category:Real algebraic geometry]] | |||
Revision as of 07:52, 6 December 2013
In abstract algebra, an ordered ring is a commutative ring 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.
Ordered rings are familiar from arithmetic. Examples include the real numbers. (The rationals and reals in fact form ordered fields.) The complex numbers 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 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 of a, denoted |a|, is defined thus:
where -a is the additive inverse of a and 0 is the additive identity element.
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
if a < b and c > 0 then ac < bc is the correct definition. For all a, b and c in R:
- If a ≤ b and 0 ≤ c, then ac ≤ bc.[1] This property is sometimes used to define ordered rings instead of the second property in the definition above.
- |ab| = |a| |b|.[2]
- An ordered ring that is not trivial is infinite.[3]
- Exactly one of the following is true: a is positive, -a is positive, or a = 0.[4] This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
- An ordered ring R has no zero divisors if and only if the positive ring elements are closed under multiplication (i.e. if a and b are positive, then so is ab).[5]
- In an ordered ring, no negative element is a square.[6] 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.
Notes
The names below refer to theorems formally verified by the IsarMathLib project. 43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.