# Definition:Ordered Integral Domain

## Definition

### Definition 1

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

 $(P \, 1)$ $:$ Closure under Ring Addition: $\displaystyle \forall a, b \in D:$ $\displaystyle \map P a \land \map P b \implies \map P {a + b}$ $(P \, 2)$ $:$ Closure under Ring Product: $\displaystyle \forall a, b \in D:$ $\displaystyle \map P a \land \map P b \implies \map P {a \times b}$ $(P \, 3)$ $:$ Trichotomy Law: $\displaystyle \forall a \in D:$ $\displaystyle \map P a \lor \map P {-a} \lor a = 0_D$ For $P \, 3$, 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 $\struct {D, +, \times}$:

 $(OID \, 1)$ $:$ $\le$ is compatible with ring addition: $\displaystyle \forall a, b, c \in D:$ $\displaystyle a \le b$ $\displaystyle \implies$ $\displaystyle \paren {a + c} \le \paren {b + c}$ $(OID \, 2)$ $:$ 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.