One-dimensional domain: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Field]] | * [[Weaker than::Field]] | ||
* [[Polynomial ring over a field]] | * [[Weaker than::Polynomial ring over a field]] | ||
* [[Ring of integers in a number field]] | * [[Weaker than::Ring of integers in a number field]] | ||
* [[Principal ideal domain]] | * [[Weaker than::Principal ideal domain]] | ||
* [[Dedekind domain]] | * [[Weaker than::Dedekind domain]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Finite-dimensional domain]] | * [[Stronger than::Finite-dimensional domain]] | ||
* [[Stronger than::Finite-dimensional ring]] | |||
Latest revision as of 17:10, 17 January 2009
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
An integral domain is termed a one-dimensional domain if it satisfies the following equivalent conditions:
- Every nonzero prime ideal in it is maximal
- It has Krull dimension at most one (note that the Krull dimension is zero iff it is a field)
Relation with other properties
Stronger properties
- Field
- Polynomial ring over a field
- Ring of integers in a number field
- Principal ideal domain
- Dedekind domain