Local ring
From Commalg
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 pointView other properties of commutative unital rings determined by the spectrum
Contents
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.