ordered ringOrderedRing2001-10-21 02:04:502006-07-22 03:34:56Definitionordered field\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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\usepackage{xypic}An \emph{ordered ring} is a commutative ring $R$ with a total ordering $\leq$ such that, for every $a,b,c \in R$:
\begin{enumerate}
\item If $a \leq b$, then $a+c \leq b+c$
\item If $a \leq b$ and $0 \leq c$, then $c \cdot a \leq c \cdot b$
\end{enumerate}
An \emph{ordered field} is an ordered ring $(R,\leq)$ where $R$ is also a field.
Examples of ordered rings include:
\begin{itemize}
\item The integers $\mathbb{Z}$, under the standard ordering $\leq$.
\item The real numbers $\mathbb{R}$ under the standard ordering.
\item 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.
\end{itemize}
Examples of rings which do not admit any ordering relation making them into an ordered ring include:
\begin{itemize}
\item The complex numbers $\mathbb{C}$.
\item The finite field $\mathbb{Z}/p\mathbb{Z}$, where $p$ is any prime.
\end{itemize}