Jacobson ring

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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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.

Definition with symbols

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Relation with other properties

Stronger properties

• Polynomial ring over a field
• Artinian ring
• Zero-dimensional ring

Property operators

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.