# Ordered integral domain

From Maths

## Contents

## Definition

An integral domain [ilmath]D[/ilmath] is said to be an *ordered integral domain*^{[1]} if it contains a subset, which we'll denote [ilmath]D^+[/ilmath] with the following properties:

- [ilmath]a,b\in D^+\implies a+b\in D^+[/ilmath] (closed under addition)
- [ilmath]a,b\in D^+\implies ab\in D^+[/ilmath] (closed under multiplication)
- [ilmath]\forall a\in D[/ilmath] exactly one of the following is true (Trichotomy law)
- [ilmath]a=0[/ilmath]
- [ilmath]a\in D^+[/ilmath]
- [ilmath]-a\in D^+[/ilmath]

Note:

- The elements of [ilmath]D^+[/ilmath] are called the
*positive elements*of [ilmath]D[/ilmath] - The non-zero elements of [ilmath]D[/ilmath] that are not in [ilmath]D^+[/ilmath] are called the
*negative elements*of [ilmath]D[/ilmath] - The [ilmath]+[/ilmath] in [ilmath]D^+[/ilmath] has nothing to do with the addition operator, it's just notation

## Examples

- [ilmath]\mathbb{Z}^+[/ilmath] is the set of positive elements of [ilmath]\mathbb{Z} [/ilmath]

## See also

## References

- ↑ Fundamentals of Abstract Algebra - An Expanded Version - Neal H. McCoy