positivity in ordered ring
Theorem.
If $(R,\le )$ is an ordered ring, then it contains a subset ${R}_{+}$ having the following :

•
${R}_{+}$ is under ring addition and, supposing that the ring contains no zero divisors, also under ring multiplication.
 •
Proof. We take $$. Let $a,b\in {R}_{+}$. Then $$, $$, and therefore we have $$, i.e. $a+b\in {R}_{+}$. If $R$ has no zerodivisors, then also $ab\ne 0$ and $$, i.e. $ab\in {R}_{+}$. Let $r$ be an arbitrary nonzero element of $R$. Then we must have either $$ or $$ (not both) because $R$ is totally ordered^{}. The latter alternative gives that $$. The both cases that either $r\in {R}_{+}$ or $r\in {R}_{+}$.
Title  positivity in ordered ring 

Canonical name  PositivityInOrderedRing 
Date of creation  20130322 14:46:40 
Last modified on  20130322 14:46:40 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  12 
Author  pahio (2872) 
Entry type  Theorem 
Classification  msc 06F25 
Classification  msc 12J15 
Classification  msc 13J25 
Related topic  PositiveCone 
Related topic  TopicEntryOnRealNumbers 