Proposition 3

Let $(R_i)_{1 \leq i \leq n}$ be a finite family of (not necessarily commutative) $A$-algebras and let $R = \prod_{i=1}^n R_i$ be their product. For an element $x = (x_i)_{1 \leq i \leq n}$ of $R$ to be integral over $A$, it is necessary and sufficient that each of the $x_i$ be integral over $A$. For $R$ to be integral over $A$, it is necessary and sufficient that each of the $R_i$ be integral over $A$.

It is obviously sufficient to prove the first assertion. The condition is necessary by Proposition 2. Conversely, if each of the $x_i$ is integral over $A$, the subalgebra $A[x_i]$ of $R_i$ is a finitely generated $A$-module and hence so is the subalgebra $\prod_{i=1}^n A[x_i]$ of $R$; as $A[x]$ is contained in this subalgebra, $x$ is integral over $A$ by Proposition 1.