Local ring

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

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Definition for noncommutative rings

Further information: Noncommutative:Local ring)

Relation with other properties

Stronger properties

• Field
• Regular local ring
• Completely primary ring
• Local Artinian ring

Weaker properties

• Semilocal ring
• Quasilocal ring

Conjunction with other properties

• Local Noetherian ring
• Local Artinian ring
• Local domain

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.