# ordered ring

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

1. 1.

If $a\leq b$, then $a+c\leq b+c$

2. 2.

If $a\leq b$ and $0\leq c$, then $c\cdot a\leq c\cdot b$

An ordered field is an ordered ring $(R,\leq)$ where $R$ is also a field.

Examples of ordered rings include:

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

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

• The polynomial ring $\mathbb{R}[x]$ in one variable over $\mathbb{R}$, under the relation $f\leq g$ 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 $\mathbb{C}$.

• 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