Let $\mathcal O$ be the ring of integers of a number field $K$ and let $M$ be an invertible $\mathcal O$-module. It means that $M$ is a finitely generated and projective $\mathcal O$-module of rank $1$.
Now let $x\in M\setminus \{0\}$ and consider the submodule $x\mathcal O$.
Why does the quotient module $M/x\mathcal O$ have finite cardinality?
I think one should use the fact that for any nonzero ideal $I\subset \mathcal O$ we have that $\mathcal O/I$ has finite cardinality. This is easy to show and you can assume it as true.
Thanks in advance.
Because it is a finitely generated torsion $\mathcal O$-module, so it is a finitely generated module over a quotient of $\mathcal O$, which is a finite ring.