Zero-dimensional ring: Difference between revisions
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==Definition== | ==Definition== | ||
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* Every [[prime ideal]] in it is [[maximal ideal|maximal]] | * Every [[prime ideal]] in it is [[maximal ideal|maximal]] | ||
* Any [[quotient ring]] of it that is an integral domain is also a field | * Any [[quotient ring]] of it that is an integral domain is also a field | ||
* The [[spectrum of a commutative unital ring|spectrum]] of the ring is a [[tps:T1 space|T1 space]] i.e. all points in the spectrum are closed | |||
==Relation with other properties== | ==Relation with other properties== | ||
===Conjunction with other properties=== | |||
* [[Zero-dimensional Noetherian ring]]: A zero-dimensional ring that is also a [[Noetherian ring]]. | |||
===Stronger properties=== | ===Stronger properties=== | ||
* [[Finite ring]] | * [[Weaker than::Finite ring]] | ||
* [[Field]] | * [[Weaker than::Field]] | ||
* [[Weaker than::Finite-dimensional algebra over a field]] | |||
* [[Weaker than::Artinian ring]] | |||
* [[Weaker than::Semisimple ring]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[ | * [[Stronger than::Jacobson ring]]: {{proofofstrictimplicationat|[[Zero-dimensional implies Jacobson]]|[[Jacobson not implies zero-dimensional]]}} | ||
* [[Noetherian ring]] | * [[Cohen-Macaulay ring]] (under the assumption that the ring is [[Noetherian ring|Noetherian]]): {{proofat|[[Zero-dimensional Noetherian implies Cohen-Macaulay]]}} | ||
* [[Stronger than::Equidimensional ring]] | |||
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Latest revision as of 17:12, 17 January 2009
This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki
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
This property of commutative unital rings is completely determined by the spectrum, viewed as an abstract topological space. The corresponding property of topological spaces is: T1 space
View other properties of commutative unital rings determined by the spectrum
Any integral domain satisfying this property of commutative unital rings, must be a field
Definition
Symbol-free definition
A commutative unital ring is termed zero-dimensional if it satisfies the following equivalent conditions:
- It has Krull dimension zero
- Every prime ideal in it is maximal
- Any quotient ring of it that is an integral domain is also a field
- The spectrum of the ring is a T1 space i.e. all points in the spectrum are closed
Relation with other properties
Conjunction with other properties
- Zero-dimensional Noetherian ring: A zero-dimensional ring that is also a Noetherian ring.
Stronger properties
Weaker properties
- Jacobson ring: For proof of the implication, refer Zero-dimensional implies Jacobson and for proof of its strictness (i.e. the reverse implication being false) refer Jacobson not implies zero-dimensional
- Cohen-Macaulay ring (under the assumption that the ring is Noetherian): For full proof, refer: Zero-dimensional Noetherian implies Cohen-Macaulay
- Equidimensional ring