Principal ideal domain: Difference between revisions
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* [[Unique factorization domain]]: {{proofofstrictimplicationat|[[PID implies UFD]]|[[UFD not implies PID]]}} | * [[Unique factorization domain]]: {{proofofstrictimplicationat|[[PID implies UFD]]|[[UFD not implies PID]]}} | ||
* [[Elementary divisor domain]] | * [[Elementary divisor domain]] | ||
===Conjunction expressions=== | |||
A ring is a principal ideal domain iff it is: | |||
* A [[principal ideal ring]] and an [[integral domain]]: This is a tautological statement | |||
* A [[Noetherian ring]] and a [[Bezout domain]]: {{further|[[Noetherian and Bezout implies principal ideal]]}} | |||
* A [[unique factorization domain]] and a [[Dedekind domain]]: {{further|[[Unique factorization and Dedekind implies principal ideal]]}} | |||
==Metaproperties== | ==Metaproperties== | ||
Revision as of 23:12, 7 January 2008
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 PID or Principal Ideal Domain if it satisfies the following equivalent conditions:
- Every ideal in it is principal, viz., it is a principal ideal ring
- Every prime ideal in it is principal
Note that the two conditions need not be equivalent when the underlying ring is not a domain.
Relation with other properties
Stronger properties
- Euclidean domain: For proof of the implication, refer Euclidean implies PID and for proof of its strictness (i.e. the reverse implication being false) refer PID not implies Euclidean
- Polynomial ring over a field
Weaker properties
- Dedekind domain: For proof of the implication, refer PID implies Dedekind and for proof of its strictness (i.e. the reverse implication being false) refer Dedekind not implies PID
- Bezout domain: For proof of the implication, refer PID implies Bezout and for proof of its strictness (i.e. the reverse implication being false) refer Bezout not implies PID
- Noetherian domain
- Unique factorization domain: For proof of the implication, refer PID implies UFD and for proof of its strictness (i.e. the reverse implication being false) refer UFD not implies PID
- Elementary divisor domain
Conjunction expressions
A ring is a principal ideal domain iff it is:
- A principal ideal ring and an integral domain: This is a tautological statement
- A Noetherian ring and a Bezout domain: Further information: Noetherian and Bezout implies principal ideal
- A unique factorization domain and a Dedekind domain: Further information: Unique factorization and Dedekind implies principal ideal
Metaproperties
Polynomial-closedness
This property of commutative unital rings is not closed under passing to the polynomial ring
The polynomial ring over a PID need not be a PID. Two examples are the polynomial ring over the integers, and the polynomial ring in two variables over a field. In fact, the polynomial ring over a ring is a PID iff that ring is a field.