Suppose that $A$ is a matrix over $\mathbb{Z}$. What does it mean to say that $A$ can be brought to Smith normal form?

Show that the structure theorem for finitely generated modules over $\mathbb{Z}$ (which you should state) follows from the existence of Smith normal forms for matrices over $\mathbb{Z}$.

Bring the matrix $\left(\begin{array}{cc}-4 & -6 \\ 2 & 2\end{array}\right)$ to Smith normal form.

Suppose that $M$ is the $\mathbb{Z}$-module with generators $e_{1}, e_{2}$, subject to the relations

$$-4 e_{1}+2 e_{2}=-6 e_{1}+2 e_{2}=0$$

Describe $M$ in terms of the structure theorem.