Principal ideal domain: Difference between revisions
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* [[Unique factorization domain]] | * [[Unique factorization domain]] | ||
* [[Elementary divisor domain]] | * [[Elementary divisor domain]] | ||
==Metaproperties== | |||
{{not poly-closed curing property}} | |||
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. | |||
Revision as of 22:55, 16 December 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 PID or Principal Ideal Domain if every ideal in it is principal.
Relation with other properties
Stronger properties
Weaker properties
- Dedekind domain
- Bezout domain
- Noetherian domain
- Unique factorization domain
- Elementary divisor domain
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.