Let $\alpha,\beta\in\mathbb C$ be algebraic integers, so there exist monic $p,q\in\mathbb Z[x]$ such that $p(\alpha)=q(\beta)=0$. It follows that $\mathbb Z[\alpha,\beta]$ is finitely-generated as a $\mathbb Z$-module.
I want to show directly that any submodule of $\mathbb Z[\alpha,\beta]$ is finitely generated.
I'm aware that the result is true in general, since $\mathbb Z$ is a PID and all submodules of a finitely generated module over a PID are finitely generated. But I'm curious if there is a particular direct way in the above case, without going through the general argument.