Jacobson ring

This article defines a property of commutative rings

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

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

Metaproperties

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.