Proposition 3

For a ring A to be a Jacobson ring, it is necessary and sufficient that, for every ideal a of A, the Jacobson radical of $A/a$ be equal to its nilradical (Chapter II, § 2, no. 6).

The Jacobson radical (resp. nilradical) of $A/a$ is the intersection of the maximal (resp. prime) ideals of $A/a$ (Algebra, Chapter VIII, § 6, no. 3, Definition 3 and Commutative Algebra, Chapter II, § 2, no. 6, Proposition 13). The stated condition means therefore that for every ideal $a$ of $A$ the intersection of the prime ideals containing $a$ is equal to the intersection of the maximal ideals containing $a$. This condition obviously holds for every ideal $a$ of $A$ if $A$ is a Jacobson ring; conversely, if it holds for every prime ideal of $A$, $A$ is a Jacobson ring by definition.