|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
|
|
|
| ``Additional more commonly known definition''
by jocaps on 2009-03-19 17:22:20 |
|
| Please add this more commonly known definition of a poring (I wrote it down in the preliminary chapter of my dissertation) more known in the field of real algebraic geometry where most rings dealt with are commutative and unitary:
Given a commutative unitary ring $A$, a subset $A^+$ of $A$ is called a \emph{partial ordering} if it has the following property.
\begin{itemize}
\item $a\in A \Rightarrow a^2 \in A$
\item $x,y\in A^+ \Rightarrow x+y,xy \in A^+$
\item $x,-x \in A^+ \Leftrightarrow x=0$
\end{itemize}
We say that a ring \emph{$A$ has a partial ordering $A^+$}, if there exist a subset $A^+\subset A$ such that $A^+$ is a partial ordering of $A$. We usually write it as a pair $(A,A^+)$ and call this a \emph{partially ordered ring} or a \emph{poring}
Remark:
If a commutative unitary ring $A$ has a partial ordering $A^+$ then set
$$\{a^2 : a\in A\} $$
is also a partial ordering of $A$. This partial ordering is, for obvious reasons, called the \emph{weakest partial ordering of $A$}. It is also known as the \emph{sum of squares of $A$}. |
| | [ reply | up ] | |
|
|
|
|
|
|
|
