Definition 1

A ring A is called a Jacobson ring if every prime ideal of A is the intersection of a family of maximal ideals.

Examples

(1) Every field is a Jacobson ring. (2) The ring $\mathbf{Z}$ is a Jacobson ring, the unique prime ideal which is not maximal (0) being the intersection of the maximal ideals $(p)$ of $\mathbf{Z}$, where $p$ runs through the set of prime numbers (cf. Proposition 4). (3) Let A be a Jacobson ring and let a be an ideal of A. Then $A/a$ is a Jacobson ring, for the ideals of $A/a$ are of the form $b/a$, where b is an ideal of A containing a and $b/a$ is prime (resp. maximal) if and only if b is.