Corollary 3

Let $A$ be a ring and $A'$ a ring containing $A$ and integral over $A$. If $\mathscr{R}'$ is the Jacobson radical of $A'$, $\mathscr{R}' \cap A$ is the Jacobson radical of $A$.

Let $32$ be the Jacobson radical of $A$. For every maximal ideal $m'$ of $A'$, $m' \cap A$ is a maximal ideal of $A$ (Proposition 1), hence $\mathscr{R} \subset m' \cap A$ and therefore $\mathscr{R} \subset \mathscr{R}' \cap A$ (Algebra, Chapter VIII, § 5, no. 3, Definition 3). Conversely, let $x \in 32' \cap A$; for every maximal ideal $m$ of $A$, there exists a prime ideal of $A'$ lying above $m$ (Theorem 1) and this ideal $m'$ is necessarily maximal (Proposition 1), hence $x \in m' \cap A = m$ and therefore $x \in 32$.