Definition:Ordered Integral Domain

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Definition 1

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

Definition 2

An ordered integral domain is an ordered ring $\struct {D, +, \times, \le}$ which is also an integral domain.

That is, it is an integral domain with an ordering $\le$ compatible with the ring structure of $\struct {D, +, \times}$:

\((\text {OID} 1)\)   $:$   $\le$ is compatible with ring addition:      \(\displaystyle \forall a, b, c \in D:\)    \(\displaystyle a \le b \)   \(\displaystyle \implies \)   \(\displaystyle \paren {a + c} \le \paren {b + c} \)             
\((\text {OID} 2)\)   $:$   Strict positivity is closed under ring product:      \(\displaystyle \forall a, b \in D:\)    \(\displaystyle 0_D \le a, 0_D \le b \)   \(\displaystyle \implies \)   \(\displaystyle 0_D \le a \times b \)             

An ordered integral domain can be denoted:

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

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

Also see

  • Results about ordered integral domains can be found here.