Jacobson 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: set of closed points in any closed subset is denseView other properties of commutative unital rings determined by the spectrum
Contents
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
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 ![]() |
---|---|---|---|
1 | prime as intersection of maximals | every prime ideal in it is an intersection of maximal ideals. | for every prime ideal ![]() ![]() ![]() ![]() ![]() ![]() |
2 | radical as intersection of maximals | every radical ideal in it is an intersection of maximal ideals | for every radical ideal ![]() ![]() ![]() ![]() ![]() ![]() |
3 | spectrum: closed points dense | in the spectrum, the set of closed points in any closed set is dense | in the spectrum ![]() ![]() ![]() ![]() |
4 | quotient: nilradical equals Jacobson ring | for every quotient ring, the nilradical equals the Jacobson radical | for every ideal ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() |
quotient-closed property of commutative unital rings | Yes | Jacobson is quotient-closed | If ![]() ![]() ![]() ![]() |
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 ![]() ![]() |
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 ![]() ![]() |
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.