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.