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

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