Unique factorization domain: Difference between revisions
(Started the page) |
No edit summary |
||
| Line 1: | Line 1: | ||
{{integral domain property}} | |||
==Definition== | ==Definition== | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
An [[integral domain]] is termed a '''unique factorization domain''' if every element can be expressed as a product of finite length of [[irreducible element]]s (possibly with multiplicity) in a manner that is unique upto the ordering of the elements. | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
| Line 21: | Line 23: | ||
* [[gcd domain]]: {{proofat|[[UFD implies gcd]]}} | * [[gcd domain]]: {{proofat|[[UFD implies gcd]]}} | ||
[[ | ==Metaproperties== | ||
{{poly-closed idp}} | |||
This is essentially what [[Gauss's lemma]] states. | |||
Revision as of 00:26, 17 April 2007
This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
Definition
Symbol-free definition
An integral domain is termed a unique factorization domain if every element can be expressed as a product of finite length of irreducible elements (possibly with multiplicity) in a manner that is unique upto the ordering of the elements.
Definition with symbols
Fill this in later
Relation with other properties
Stronger properties
Weaker properties
- Normal domain: For full proof, refer: UFD implies normal
- gcd domain: For full proof, refer: UFD implies gcd
Metaproperties
Polynomial-closedness
This property of integral domains is closed under taking polynomials, i.e., whenever an integral domain has this property, so does the polynomial ring in one variable over it.
View other polynomial-closed properties of integral domains OR view polynomial-closed properties of commutative unital rings
This is essentially what Gauss's lemma states.