Unique factorization domain: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Normal domain]]: {{proofat|[[ | * [[Normal domain]]: {{proofat|[[Unique factorization implies normal]]}} | ||
* [[gcd domain]]: {{proofat|[[UFD implies gcd]]}} | * [[gcd domain]]: {{proofat|[[UFD implies gcd]]}} | ||
Revision as of 08:49, 8 August 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
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Relation with other properties
Stronger properties
Weaker properties
- Normal domain: For full proof, refer: Unique factorization 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.