Exercise 8, § 3

Let $A$ be a ring. For $A$ to be a Jacobson ring, it is necessary and sufficient that, for every maximal ideal $m'$ of the polynomial ring $A[X]$, $m' \cap A$ is a maximal ideal of $A$. (Note that if an integral domain $B$ admits a Jacobson radical $\mathfrak{R} \neq 0$ and if $b \in \mathfrak{R}$, $b \neq 0$, the intersection of $B$ and a maximal ideal of $B[X]$ containing $1 + bX$ cannot contain $b$.)