Ordered integral domain

Definition

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

1. [ilmath]a,b\in D^+\implies a+b\in D^+[/ilmath] (closed under addition)
2. [ilmath]a,b\in D^+\implies ab\in D^+[/ilmath] (closed under multiplication)
3. [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]