Local ring: Difference between revisions
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{{curing property}} | {{curing property}} | ||
==Definition | {{spectrum-determined curing property|topological space with exactly one closed point}} | ||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[commutative unital ring]] is termed a '''local ring''' if it has a unique [[maximal ideal]] | A [[commutative unital ring]] is termed a '''local ring''' if it satisfies the following equivalent conditions: | ||
* It has a unique [[maximal ideal]] | |||
* There is exactly one homomorphism from the commutative unital ring whose image is a [[field]] | |||
* There is exactly one closed point in the [[spectrum]] (corresponding to the unique maximal ideal) | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
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==Definition for noncommutative rings== | ==Definition for noncommutative rings== | ||
{{further|[[Local ring | {{further|[[Noncommutative:Local ring)]]}} | ||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Field]] | * [[Field]] | ||
* [[Regular local ring]] | * [[Regular local ring]] | ||
* [[Completely primary ring]] | |||
* [[Local Artinian ring]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
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* [[Local domain]] | * [[Local domain]] | ||
[[ | ===Analogues in other algebraic structures=== | ||
* [[Groupprops:One-headed group|One-headed group]] in group theory, is a group that has a proper normal subgroup that contains every proper normal subgroup. | |||
* [[Noncommutative:Local ring|Local ring]] in noncommutative algebra. | |||
Latest revision as of 16:37, 20 January 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
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: topological space with exactly one closed point
View other properties of commutative unital rings determined by the spectrum
Definition
Symbol-free definition
A commutative unital ring is termed a local ring if it satisfies the following equivalent conditions:
- It has a unique maximal ideal
- There is exactly one homomorphism from the commutative unital ring whose image is a field
- There is exactly one closed point in the spectrum (corresponding to the unique maximal ideal)
Definition with symbols
Fill this in later
Definition for noncommutative rings
Further information: Noncommutative:Local ring)
Relation with other properties
Stronger properties
Weaker properties
Conjunction with other properties
Analogues in other algebraic structures
- One-headed group in group theory, is a group that has a proper normal subgroup that contains every proper normal subgroup.
- Local ring in noncommutative algebra.