Ordered integral domain

Definition

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

1. $a,b\in D^+\implies a+b\in D^+$ (closed under addition)
2. $a,b\in D^+\implies ab\in D^+$ (closed under multiplication)
3. $\forall a\in D^+$ exactly one of the following is true (Trichotomy law)
   • $a=0$
   • $a\in D^+$
   • $-a\in D^+$

Note:

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

Examples

• $\mathbb{Z}^+$ is the set of positive elements of $\mathbb{Z}$