Definition:Ordered Integral Domain/Definition 1

Definition

An ordered integral domain is an integral domain $\struct {D, +, \times}$ which has a strict positivity property $P$:

$(\text P 1)$	$:$	Closure under Ring Addition:	$\ds \forall a, b \in D:$	$\ds \map P a \land \map P b \implies \map P {a + b} $
$(\text P 2)$	$:$	Closure under Ring Product:	$\ds \forall a, b \in D:$	$\ds \map P a \land \map P b \implies \map P {a \times b} $
$(\text P 3)$	$:$	Trichotomy Law:	$\ds \forall a \in D:$	$\ds \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} $
		For $\text P 3$, exactly one condition applies for all $a \in D$.

An ordered integral domain can be denoted:

$\struct {D, +, \times \le}$

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order

\((\text P 1)\)	$:$	Closure under Ring Addition:	\(\ds \forall a, b \in D:\)	\(\ds \map P a \land \map P b \implies \map P {a + b} \)
\((\text P 2)\)	$:$	Closure under Ring Product:	\(\ds \forall a, b \in D:\)	\(\ds \map P a \land \map P b \implies \map P {a \times b} \)
\((\text P 3)\)	$:$	Trichotomy Law:	\(\ds \forall a \in D:\)	\(\ds \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} \)
		For $\text P 3$, exactly one condition applies for all $a \in D$.