Smith normal form and basis for the image of a module homomorphism Let's say
$$\varphi: \mathbb{Z}^n \to \mathbb{Z}^m$$
is a $\mathbb{Z}$-linear mapping and $A$ is the transformation matrix of $\varphi$ and
$$SAT = Q$$
where $Q$ is the Smith normal form of $A$ with $S$ and $T$ both invertible over the Ring $\mathbb{Z}$.
I want to prove that the columns of $AT$ are a basis of $im(\varphi)$ but I cannot come up with a solution.

The image of $\varphi$ consists of the vectors $Av$ for $v\in\Bbb Z^n$. As $T$ is invertible over $\Bbb Z$, the $ATw$ for $w\in\Bbb Z^n$ are the same as the $Av$
for $v\in\Bbb Z^n$. So this image is the $\Bbb Z$-span of the columns of $AT$
(as well as the $\Bbb Z$-span of the columns of $T$).
