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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* [[Field]] | * [[Field]] | ||
* [[Regular local ring]] | * [[Regular local ring]] | ||
* [[Completely primary ring]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
Revision as of 21:48, 20 January 2008
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: Local ring (noncommutative rings)