Difference between revisions of "Jacobson ring"

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===Symbol-free definition===
 
===Symbol-free definition===
  
A [[commutative unital ring]] is termed a '''Jacobson ring''' or a '''Hilbert ring''' if it satisfies the following equivalent conditions:
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The following are equivalent definitions of Jacobson ring.
  
* Every [[prime ideal]] in it is an intersection of [[maximal ideal]]s
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{| class="sortable" border="1"
* Every [[radical ideal]] in it is an intersection of maximal ideals
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! No. !! Shorthand !! A commutative unital ring is termed a Jacobson ring or Hilbert ring if ... !! A commutative unital ring <math>R</math> is termed a Jacobson ring or Hilbert ring if ...
* In the [[spectrum of a commutative unital ring|spectrum]], the set of closed points in any closed set is dense
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|-
* For every [[quotient ring]], the [[nilradical]] equals the [[Jacobson radical]]
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| 1 || prime as intersection of maximals || every [[defining ingredient::prime ideal]] in it is an intersection of [[defining ingredient::maximal ideal]]s. || for every prime ideal <math>P</math> of <math>R</math>, <math>P = \bigcap_{i \in I} M_i</math> where <math>M_i, i \in I</math> is the set of all maximal ideals of <math>R</math> containing <math>P</math>.
* For any [[prime ideal]], if the quotient contains an element at which its localization is a field, then the quotient is itself a field
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|-
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| 2 || radical as intersection of maximals || every [[radical ideal]] in it is an intersection of maximal ideals || for every radical ideal <math>J</math> of <math>R</math>, <math>J = \bigcap_{i \in I} M_i</math> where <math>M_i, i \in I</math> is the set of all maximal ideals of <math>R</math> containing <math>J</math>.
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|-
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| 3 || spectrum: closed points dense || in the [[spectrum of a commutative unital ring|spectrum]], the set of closed points in any closed set is dense || in the spectrum <math>\operatorname{Spec}(R)</math>, any closed subset <math>C</math> has the property that the set of closed points in <math>C</math> is a dense subset of <math>C</math>.
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|-
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| 4 || quotient: nilradical equals Jacobson ring || for every [[quotient ring]], the [[defining ingredient::nilradical]] equals the [[defining ingredient::Jacobson radical]] || for every ideal <math>I</math>, the quotient ring <math>R/I</math> has the property that the nilradical of <math>R/I</math> (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).
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|-
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| 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 <math>P</math> and any <math>x \in R/P</math>, consider the [[localization at a multiplicative subset|localization]] of <math>R/P</math> at the set of powers of <math>x</math>. If the localization is a field, then <math>R/P</math> must itself be a field.
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|}
  
 
===Definition with symbols===
 
===Definition with symbols===
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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 ideal]]s.
 
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 ideal]]s.
  
The equivalence with the fourth condition is termed [[Rabinowitch's trick]]
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The equivalence with the fourth condition is termed [[Rabinowitch's trick]].
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==Relation with other properties==
 
==Relation with other properties==

Revision as of 15:47, 18 July 2013

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 dense


View 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 R 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 P of R, P = \bigcap_{i \in I} M_i where M_i, i \in I is the set of all maximal ideals of R containing P.
2 radical as intersection of maximals every radical ideal in it is an intersection of maximal ideals for every radical ideal J of R, J = \bigcap_{i \in I} M_i where M_i, i \in I is the set of all maximal ideals of R containing J.
3 spectrum: closed points dense in the spectrum, the set of closed points in any closed set is dense in the spectrum \operatorname{Spec}(R), any closed subset C has the property that the set of closed points in C is a dense subset of C.
4 quotient: nilradical equals Jacobson ring for every quotient ring, the nilradical equals the Jacobson radical for every ideal I, the quotient ring R/I has the property that the nilradical of R/I (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 P and any x \in R/P, consider the localization of R/P at the set of powers of x. If the localization is a field, then R/P must itself be a field.

Definition with symbols

Fill this in later

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.


Relation with other properties

Stronger properties

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 property


View 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 R is a commutative unital ring satisfying the property, so is R[x]


View other polynomial-closed properties of commutative unital rings

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

For full proof, refer: Jacobson is polynomial-closed