PlanetMath: ordered integral domain with well-ordered positive elements

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ordered integral domain with well-ordered positive elements

(Theorem)

Theorem 1 If $(R,\,\leq)$ is an ordered integral domain and if the set $R_+ = \{r\in R: \,\,0 < r\}$ of its positive elements is well-ordered, then

and

can be expressed as sets of multiples of the unity as follows:

The theorem may be interpreted so that such an integral domain is isomorphic with the ordered ring $\mathbb{Z}$ of rational integers.

"ordered integral domain with well-ordered positive elements" is owned by Wkbj79. [ owner history (1) ]

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Also defines:

positive element

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AMS MSC:	13J25 (Commutative rings and algebras :: Topological rings and modules :: Ordered rings)
	12J15 (Field theory and polynomials :: Topological fields :: Ordered fields)
	06F25 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered rings, algebras, modules)

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