ordered ring
An ordered ring is a commutative ring $R$ with a total ordering $\le $ such that, for every $a,b,c\in R$:

1.
If $a\le b$, then $a+c\le b+c$

2.
If $a\le b$ and $0\le c$, then $c\cdot a\le c\cdot b$
An ordered field is an ordered ring $(R,\le )$ where $R$ is also a field.
Examples of ordered rings include:

•
The integers $\mathbb{Z}$, under the standard ordering^{} $\le $.

•
The real numbers $\mathbb{R}$ under the standard ordering.

•
The polynomial ring $\mathbb{R}[x]$ in one variable over $\mathbb{R}$, under the relation $f\le g$ if and only if $gf$ has nonnegative leading coefficient.
Examples of rings which do not admit any ordering relation making them into an ordered ring include:

•
The complex numbers^{} $\u2102$.

•
The finite field^{} $\mathbb{Z}/p\mathbb{Z}$, where $p$ is any prime.
Title  ordered ring 
Canonical name  OrderedRing 
Date of creation  20130322 11:52:06 
Last modified on  20130322 11:52:06 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  13 
Author  djao (24) 
Entry type  Definition 
Classification  msc 06F25 
Classification  msc 12J15 
Classification  msc 13J25 
Classification  msc 11D41 
Related topic  TotalOrder 
Related topic  OrderingRelation 
Defines  ordered field 