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 $T$).