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ordered integral domain with wellordered positive elements
Theorem.
If $(R,\,\leq)$ is an ordered integral domain and if the set $R_{+}=\{r\in R:\,\,0<r\}$ of its positive elements is wellordered, 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.
Defines:
positive element
Related:
TotalOrder, OrderedRing
Type of Math Object:
Theorem
Major Section:
Reference
Parent:
Groups audience:
Mathematics Subject Classification
06F25 no label found12J15 no label found13J25 no label found Forums
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