Jacobson ring

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.

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.

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.