Jacobson ring

Remarks

Example: $\Z$ is a Jacobson ring. A nonzero prime ideal $p\Z$ is maximal, and the prime $0$ equals $\bigcap_p p\Z$, because a nonzero integer has only finitely many prime divisors.

Non-example: the localization $\Z_{(p)}$ is not a Jacobson ring. Its only maximal ideal is $p\Z_{(p)}$, so the intersection of the maximal ideals containing the prime $0$ is $p\Z_{(p)} \ne 0$.

Equivalent forms: $R$ is Jacobson if and only if every radical ideal is an intersection of maximal ideals. It is also equivalent that for every ideal $I$, the nilradical of $R/I$ equals its Jacobson radical. Some texts use the name Hilbert ring.