ordered integral domain with wellordered positive elements
Theorem.
If $(R,\le )$ is an ordered (http://planetmath.org/OrderedRing) integral domain^{} and if the set $$ of its positive elements^{} (http://planetmath.org/PositivityInOrderedRing) is wellordered, then $R$ and ${R}_{+}$ can be expressed as sets of multiples of the unity as follows:

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$R=\{m\cdot 1:m\in \mathbb{Z}\}$,

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${R}_{+}=\{n\cdot 1:n\in {\mathbb{Z}}_{+}\}$.
The theorem may be interpreted so that such an integral domain is isomorphic^{} with the ordered ring $\mathbb{Z}$ of rational integers.
Title  ordered integral domain with wellordered positive elements 

Canonical name  OrderedIntegralDomainWithWellorderedPositiveElements 
Date of creation  20130322 14:46:43 
Last modified on  20130322 14:46:43 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  11 
Author  Wkbj79 (1863) 
Entry type  Theorem 
Classification  msc 06F25 
Classification  msc 12J15 
Classification  msc 13J25 
Related topic  TotalOrder 
Related topic  OrderedRing 
Defines  positive element 