Difference between revisions of "Ordered integral domain"
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| − | {{ | + | ==Definition== |
| + | An [[Integral domain|integral domain]] {{M|D}} is said to be an ''ordered integral domain''<ref name="FOAA">Fundamentals of Abstract Algebra - An Expanded Version - Neal H. McCoy</ref> if it contains a subset, which we'll denote {{M|D^+}} with the following properties: | ||
| + | # {{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|\forall a\in D^+}} exactly one of the following is true ([[Trichotomy law]]) | ||
| + | #* {{M|1=a=0}} | ||
| + | #* {{M|a\in D^+}} | ||
| + | #* {{M|-a\in D^+}} | ||
| + | Note: | ||
| + | * The elements of {{M|D^+}} are called the ''positive 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 | ||
| + | ==Examples== | ||
| + | * {{M|\mathbb{Z}^+}} is the set of positive elements of {{M|\mathbb{Z} }} | ||
| + | ==References== | ||
| + | <references/> | ||
| + | {{Definition|Abstract Algebra}} | ||
Revision as of 05:46, 9 June 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:
- [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]
References
- ↑ Fundamentals of Abstract Algebra - An Expanded Version - Neal H. McCoy
