Classification of $M_{n}(\mathbb{Z})$-submodules of $\mathbb{Z}^n$

Question

I recently saw this question and wanted to generalize it to $\mathbb{Z}^n$ and matrices with integer coefficients. I think $k\mathbb{Z}^n$ is a submodule for any integer $k$. Is there a nice way to generalize this?

Answer 1

All submodules of $\mathbb{Z}^n$ are of the form $k\mathbb{Z}^n$. Namely, if $S$ is a nonzero submodule then $S = d\mathbb{Z}^n$, where $d$ is the gcd of all integers which appear as entries of elements of $S$. This can be seen by repeated application of Bézout's identity.