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 $ℝ$ under the standard ordering.

• The polynomial ring $ℝ[x]$ in one variable over $ℝ$, under the relation $f\le g$ if and only if $g-f$ has nonnegative leading coefficient.

Examples of rings which do not admit any ordering relation making them into an ordered ring include:

• The complex numbers $ℂ$.

• The finite field $\mathbb{Z}/p\mathbb{Z}$, where $p$ is any prime.

Title	ordered ring
Canonical name	OrderedRing
Date of creation	2013-03-22 11:52:06
Last modified on	2013-03-22 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