Proposition 1

Let $A$ be a ring, $R$ an algebra over $A$ (not necessarily commutative) and $x$ an element of $R$. For $x$ to be integral over $A$, it is necessary and sufficient that $A[x]$ be contained in a subalgebra $R'$ of $R$ which is a finitely generated $A$-module.

The condition is obviously necessary by virtue of property (E_{II}); it is also sufficient by virtue of (E_{III}), for $R'$ is a faithful $A[x]$-module (since it contains the unit element of $R$).