# Jacobson 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 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: set of closed points in any closed subset is denseView other properties of commutative unital rings determined by the spectrum

## History

### Origin of the term

The term **Jacobson ring** was used by Krull in honour of Jacobson, who studied intersections of maximal ideals.

### Alternative terminology

The term **Hilbert ring** or **Hilbertian ring** is also used because such rings are closely related to the Hilbert nullstellensatz.

## Definition

### Symbol-free definition

The following are equivalent definitions of Jacobson ring.

No. | Shorthand | A commutative unital ring is termed a Jacobson ring or Hilbert ring if ... | A commutative unital ring is termed a Jacobson ring or Hilbert ring if ... |
---|---|---|---|

1 | prime as intersection of maximals | every prime ideal in it is an intersection of maximal ideals. | for every prime ideal of , where is the set of all maximal ideals of containing . |

2 | radical as intersection of maximals | every radical ideal in it is an intersection of maximal ideals | for every radical ideal of , where is the set of all maximal ideals of containing . |

3 | spectrum: closed points dense | in the spectrum, the set of closed points in any closed set is dense | in the spectrum , any closed subset has the property that the set of closed points in is a dense subset of . |

4 | quotient: nilradical equals Jacobson ring | for every quotient ring, the nilradical equals the Jacobson radical | for every ideal , the quotient ring has the property that the nilradical of (i.e., the set of nilpotent elements, or equivalently, the intersection of all prime ideals) equals the Jacobson radical (the set of elements such that 1 + any multiple of the element is invertible, or equivalently, the intersection of all maximal ideals). |

5 | localization a field implies a field | for any prime ideal, if the quotient contains an element at which its localization is a field, then the quotient is itself a field. | for any prime ideal and any , consider the localization of at the set of powers of . If the localization is a field, then must itself be a field. |

### Equivalence of definitions

The equivalence of the first three definition follows from the definitions of the terms involved. In particular, it uses the fact that in any commutative unital ring, any radical ideal is an intersection of prime ideals.

The equivalence with the fourth condition is termed Rabinowitch's trick.

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

polynomial-closed property of commutative unital rings | Yes | Jacobson is polynomial-closed | Suppose is a Jacobson ring. Then, the polynomial ring is also a Jacobson ring. |

quotient-closed property of commutative unital rings | Yes | Jacobson is quotient-closed | If is a Jacobson ring and is an ideal in , then the quotient ring is also a Jacobson ring. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

polynomial ring over a field | polynomial ring of the form where is a field. | Template:Intermeidate notions short | ||

field | click here | |||

Artinian ring | click here | |||

zero-dimensional ring | every prime ideal is a maximal ideal | click here | ||

multivariate polynomial ring over a field | of the form where is a field. | click here |

### Opposite properties

A local domain that is not a field is not Jacobson. More generally, any local ring that has prime ideals other than the maximal ideal is not Jacobson.

## Metaproperties

### Closure under taking quotient rings

This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this propertyView other quotient-closed properties of commutative unital rings

A quotient of a Jacobson ring by any ideal is also a Jacobson ring. This is clear, for instance, if we use the characterization in terms of every quotient ring having equal nilradical and Jacobson radical.*For full proof, refer: Jacobson is quotient-closed*

### Closure under taking the polynomial ring

This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if is a commutative unital ring satisfying the property, so isView other polynomial-closed properties of commutative unital rings

If is a Jacobson ring, so is the polynomial ring . This is an important observation that forms part of the proof of the Hilbert nullstellensatz (where the starting ring, is a field and hence clearly a Jacobson ring).

*For full proof, refer: Jacobson is polynomial-closed*