# 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 ringsVIEW 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.