Principal ideal ring: Difference between revisions
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{{curing property}} | |||
==Definition== | ==Definition== | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[commutative unital ring]] | A [[commutative unital ring]] is termed a '''principal ideal ring''' if every [[ideal]] in it is [[principal ideal|principal]], that is, if every ideal is generated by a single element. | ||
===Definition with symbols=== | ===Definition with symbols=== | ||
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===Conjunction with other properties=== | ===Conjunction with other properties=== | ||
* [[Principal ideal domain]] is a principal ideal ring which is also an [[integral domain]] | * [[Weaker than::Principal ideal domain]] is a principal ideal ring which is also an [[integral domain]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Bezout ring]] | * [[Stronger than::Bezout ring]] | ||
* [[Noetherian ring]] | * [[Stronger than::Noetherian ring]] | ||
* [[Stronger than::One-dimensional ring]]: {{proofat|[[Principal ideal ring implies one-dimensional]]}} | |||
[[ | ==Metaproperties== | ||
{{Q-closed curing property}} | |||
{{localization-closed curing property}} | |||
{{proofat|[[Principal ideal ring is localization-closed]]}} | |||
Latest revision as of 23:24, 8 February 2009
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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
A commutative unital ring is termed a principal ideal ring if every ideal in it is principal, that is, if every ideal is generated by a single element.
Definition with symbols
Fill this in later
Relation with other properties
Conjunction with other properties
- Principal ideal domain is a principal ideal ring which is also an integral domain
Weaker properties
- Bezout ring
- Noetherian ring
- One-dimensional ring: For full proof, refer: Principal ideal ring implies one-dimensional
Metaproperties
Closure under taking quotient rings
This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this property
View other quotient-closed properties of commutative unital rings
Closure under taking localizations
This property of commutative unital rings is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.
View other localization-closed properties of commutative unital rings
For full proof, refer: Principal ideal ring is localization-closed