ordered integral domain with well-ordered 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 well-ordered, then $R$ and $R_{+}$ can be expressed as sets of multiples of the unity as follows:

• $R=\{m\cdot 1:m\in \mathbb{Z}\}$,

• $R_{+}=\{n\cdot 1:n\in {\mathbb{Z}}_{+}\}$.

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 well-ordered positive elements
Canonical name	OrderedIntegralDomainWithWellorderedPositiveElements
Date of creation	2013-03-22 14:46:43
Last modified on	2013-03-22 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