Proposition 6

Let $A$ be a ring, $B$ a (commutative)A-algebra and $C$ a (not necessarily commutative)B-algebra. If $B$ is integral over $A$ and $C$ is integral over $B$, then $C$ is integral over $A$.

It is sufficient to verify that every $x \in C$ is integral over $A$. By hypothesis there exists a monic polynomial $X^n + b_1 X^{n-1} + \cdots + b$, with coefficients in $B$ with $x$ as a root; then $x$ is integral over $B' = A[b_1, \ldots, b_n]$ and $B'[x]$ is therefore a finitely generated B-module. But as $B$ is integral over $A$, $B'$ is a finitely generated A-module (Proposition 4); we conclude that B'[x] is also a finitely generated A-module (Algebra, Chapter 11, § 1, no. 13, Proposition 25) and therefore x is integral over A.