= Solution
The <Smith normal form> of an integer $m\times n$ <matrix> $A$ is a diagonal <matrix>
$$
D=\operatorname{diag}(d_1,\ldots,d_r,0,\ldots,0),
\qquad d_i>0,
\qquad d_i\mid d_{i+1},
$$
for which
$$
UAV=D
$$
with $U\in GL_m(\mathbb Z)$ and $V\in GL_n(\mathbb Z)$. It exists and is unique.
The <structure theorem for finitely generated modules over a principal ideal domain>, specialized to $\mathbb Z$, says that every finitely generated abelian <group> has a unique invariant-factor decomposition
$$
M\cong\mathbb Z^s\oplus
\mathbb Z/d_1\mathbb Z\oplus\cdots\oplus
\mathbb Z/d_r\mathbb Z,
\qquad
1<d_1\mid\cdots\mid d_r.
$$
Equivalently, its finite part is a direct sum of cyclic <groups> of prime-power order.
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