It is a very well known fact that Smith Normal Form has proven useful when dealing with the development of the structure theorem of finitely generated abelian groups. In this context, there is an approach that takes advantage of the next result, which indeed is a very particular case of a much more general theorem related with a special kind of rings.
If $A$ is a $m\times n$ matrix with integer coefficients, then there exist two matrices $P$ of size $m\times m$ and $Q$ of size $n\times n$, both having integer entries and $\det =\pm 1$, such that $PAQ$ is a diagonal matrix with diagonal entries $d_1,d_2,\ldots,d_k$ ($k<\min(m,n)$) such that $d_1\mid d_2\mid \ldots\mid d_k$, and each $d_i$ is a positive integer.
Furthermore, $d_1\mid \ldots\mid d_k$ are unique.
I have no problem with the proof of the "existence" part of the last theorem. However, I can't manage to give a proof of the "uniqueness" part; at most, I can only show that if $d'_1\mid\ldots\mid d'_{\ell}$ have the same property, then $d'_1 = d_1$ (they are both the gcd of the entries of $A$).
The idea is not to give a proof using structure theorems, but only any kind of "very elemental" proof (dealing, if possible, just with $\mathbb{Z}$ properties, and not talking about general rings/modules).
Best Answer
See here (page 76). The main idea is this:
For each k, the GCD of all the determinants of $k \times k$ submatrices of $A$ (with possibly different sets of indices for columns and rows) is preserved under multiplying by integer matrices with $\det =\pm 1$. These invariants equal $d_1 \cdot \ldots \cdot d_k$ in the smith form for each $k$, determining it uniquely.