Difference between revisions of "Ordered integral domain"
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# {{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 | + | # {{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} }} | ||
| + | |||
| + | ==See also== | ||
| + | * [[Ring]] | ||
| + | * [[Group]] | ||
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Abstract Algebra}} | {{Definition|Abstract Algebra}} | ||
Latest revision as of 16:10, 23 August 2015
Contents
[hide]Definition
An integral domain D is said to be an ordered integral domain[1] if it contains a subset, which we'll denote D+ with the following properties:
- a,b∈D+⟹a+b∈D+ (closed under addition)
- a,b∈D+⟹ab∈D+ (closed under multiplication)
- ∀a∈D exactly one of the following is true (Trichotomy law)
- a=0
- a∈D+
- −a∈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
- Z+ is the set of positive elements of Z
See also
References
- Jump up ↑ Fundamentals of Abstract Algebra - An Expanded Version - Neal H. McCoy
