Definition:Ordered Integral Domain

Definition
An ordered integral domain is an integral domain $\left({D, +, \times}\right)$ with a property $P$ such that:


 * $(1): \quad \forall a, b \in D: P \left({a}\right) \land P \left({b}\right) \implies P \left({a + b}\right)$


 * $(2): \quad \forall a, b \in D: P \left({a}\right) \land P \left({b}\right) \implies P \left({a \times b}\right)$


 * $(3): \quad \forall a \in D: P \left({a}\right) \lor P \left({-a}\right) \lor a = 0_D$

For condition $(3)$, exactly one of the conditions applies for every element of $D$.

An ordered integral domain can be denoted:
 * $\left({D, +, \times \le}\right)$

where $\le$ is the ordering induced by the positivity property.

Also see

 * From Positivity Property induces Total Ordering it can be seen that this definition is equivalent to that of a totally ordered ring.


 * Trichotomy Law (in the context of an ordering). Note that the two statements of the trichotomy law are ultimately equivalent, but the one given on this page is ultimately more general and fundamental.