Ordered ring

<h2 id="examples">Examples</h2>
Ordered rings are familiar from <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a>. Examples include the <a href="/facts/Integer/PUwwrawx">integers</a>, the <a href="/facts/Rational_number/VJQmyY05">rationals</a> and the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> (The rationals and reals in fact form <a href="/facts/Ordered_field/an0prSgG">ordered fields</a>.) The <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and i.

<h2 id="positive-elements">Positive elements</h2>
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.

<h2 id="absolute-value">Absolute value</h2>
If 
 
 
 
 a
 
 
 {\displaystyle a}
 
 is an element of an ordered ring R, then the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> of 
 
 
 
 a
 
 
 {\displaystyle a}
 
, denoted 
 
 
 
 
 |
 
 a
 
 |
 
 
 
 {\displaystyle |a|}
 
, is defined thus:

|
        
        a
        
          |
        
        :=
        
          
            {
            
              
                
                  a
                  ,
                
                
                  
                    
                      if 
                    
                  
                  0
                  ≤
                  a
                  ,
                
              
              
                
                  −
                  a
                  ,
                
                
                  
                    
                      otherwise
                    
                  
                  ,
                
              
            
            
          
        
      
    
    {\displaystyle |a|:={\begin{cases}a,&{\mbox{if }}0\leq a,\\-a,&{\mbox{otherwise}},\end{cases}}}

where 
 
 
 
 −
 a
 
 
 {\displaystyle -a}
 
 is the <a href="/facts/Additive_inverse/n8gKUmCf">additive inverse</a> of 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 0 is the additive <a href="/facts/Identity_element/9nss3xHY">identity element</a>.

<h2 id="discrete-ordered-rings">Discrete ordered rings</h2>
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.

<h2 id="basic-properties">Basic properties</h2>
For all a, b and c in R:

<ul><li>If a ≤ b and 0 ≤ c, then ac ≤ bc.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> This property is sometimes used to define ordered rings instead of the second property in the definition above.</li>
<li>|ab| = |a| |b|.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>An ordered ring that is not <a href="/facts/Trivial_ring/trFjtDJ1">trivial</a> is infinite.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>Exactly one of the following is true: a is positive, −a is positive, or a = 0.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> This property follows from the fact that ordered rings are <a href="/facts/Abelian_group/FfWwmx4R">abelian</a>, <a href="/facts/Linearly_ordered_group/APXGv4UI">linearly ordered groups</a> with respect to addition.</li>
<li>In an ordered ring, no negative element is a square:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Firstly, 0 is square. Now if a ≠ 0 and a = b2 then b ≠ 0 and a = (−b)2; as either b or −b is positive, a must be nonnegative.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Ordered_field/an0prSgG">Ordered field</a> – Algebraic object with an ordered structure</li>
<li><a href="/facts/Ordered_group/xoLNpEqH">Ordered group</a> – Group with a compatible partial orderPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Ordered_topological_vector_space/N1fvN8th">Ordered topological vector space</a></li>
<li><a href="/facts/Ordered_vector_space/yN9VfKMa">Ordered vector space</a> – Vector space with a partial order</li>
<li><a href="/facts/Partially_ordered_ring/s0mAmlFv">Partially ordered ring</a> – Ring with a compatible partial order</li>
<li><a href="/facts/Partially_ordered_space/06RwXYXo">Partially ordered space</a> – Partially ordered topological space</li>
<li><a href="/facts/Riesz_space/FX7Mwfxe">Riesz space</a> – Partially ordered vector space, ordered as a lattice, also called vector lattice</li>
<li><a href="/facts/Semiring/7cSvWXVQ">Ordered semirings</a></li></ul>
<h2 id="notes">Notes</h2>
The list below includes references to theorems formally verified by the <a href="http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf">IsarMathLib</a> project.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Lam, T. Y. (1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, American Mathematical Society, ISBN 0-8218-0702-1, Zbl 0516.12001 <a href="0-8218-0702-1" target="_blank">0-8218-0702-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">*Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2nd ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439, Zbl 0980.16001 <a href="0-387-95183-0" target="_blank">0-387-95183-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">OrdRing_ZF_1_L9 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">OrdRing_ZF_2_L5 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">ord_ring_infinite <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">OrdRing_ZF_3_L2, see also OrdGroup_decomp <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">OrdRing_ZF_1_L12 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Ordered ring open-in-new

Ordered ring