Algebraic integer module criterion

An element $x$ is integral over $\mathbb{Z}$ exactly when it acts stably on some faithful nonzero finitely generated $\mathbb{Z}$-module containing one.

A ring generated over $\mathbb{Z}$ by finitely many algebraic integers is a finitely generated $\mathbb{Z}$-module.

If $B$ is integral over $A$ and an element is integral over $B$, then it is integral over $A$.