Jacobson ring: Difference between revisions
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{{poly-closed | {{poly-closed curing property}} | ||
If <math>R</math> is a Jacobson ring, so is the [[polynomial ring]] <math>R[x]</math>. This is an important observation that forms part of the proof of the [[Hilbert nullstellensatz]] (where the starting ring, <math>R</math> is a field and hence clearly a Jacobson ring). | If <math>R</math> is a Jacobson ring, so is the [[polynomial ring]] <math>R[x]</math>. This is an important observation that forms part of the proof of the [[Hilbert nullstellensatz]] (where the starting ring, <math>R</math> is a field and hence clearly a Jacobson ring). | ||
Revision as of 11:57, 7 August 2007
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
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
A commutative unital ring is termed a Jacobson ring or a Hilbert ring if it satisfies the following equivalent conditions:
- Every prime ideal in it is an intersection of maximal ideals
- Every radical ideal in it is an intersection of maximal ideals
- In the spectrum, the set of closed points in any closed set is dense
Definition with symbols
Fill this in later
Relation with other properties
Stronger properties
Metaproperties
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 is
View 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).