Ordered integral domain

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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:

  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]


  • 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


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

See also


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