Difference between revisions of "Ordered integral domain"

From Maths
Jump to: navigation, search
m
m (Definition: Typo)
 
Line 3: Line 3:
 
# {{M|a,b\in D^+\implies a+b\in D^+}} (closed under addition)
 
# {{M|a,b\in D^+\implies a+b\in D^+}} (closed under addition)
 
# {{M|a,b\in D^+\implies ab\in D^+}} (closed under multiplication)
 
# {{M|a,b\in D^+\implies ab\in D^+}} (closed under multiplication)
# {{M|\forall a\in D^+}} exactly one of the following is true ([[Trichotomy law]])
+
# {{M|\forall a\in D}} exactly one of the following is true ([[Trichotomy law]])
 
#* {{M|1=a=0}}
 
#* {{M|1=a=0}}
 
#* {{M|a\in D^+}}
 
#* {{M|a\in D^+}}
Line 11: Line 11:
 
* The non-zero elements of {{M|D}} that are not in {{M|D^+}} are called the ''negative elements'' of {{M|D}}
 
* The non-zero elements of {{M|D}} that are not in {{M|D^+}} are called the ''negative elements'' of {{M|D}}
 
* The {{M|+}} in {{M|D^+}} has nothing to do with the addition operator, it's just notation
 
* The {{M|+}} in {{M|D^+}} has nothing to do with the addition operator, it's just notation
 +
 
==Examples==
 
==Examples==
 
* {{M|\mathbb{Z}^+}} is the set of positive elements of {{M|\mathbb{Z} }}
 
* {{M|\mathbb{Z}^+}} is the set of positive elements of {{M|\mathbb{Z} }}

Latest revision as of 16:10, 23 August 2015

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:

  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]

See also

References

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