Let \(R\) be a PID and \(A\in \Mat(m\times n; R)\). There are invertible matrices \(P\) and \(Q\) over \(R\) such that \(\SNF(A) = PAQ\) is diagonal with diagonal entries \(a_1\divides a_2\divides\cdots\divides a_r\) followed by zeros, and the \(a_i\) are unique up to units.
The diagonal entries are \(a_i = d_i/d_{i-1}\), where \(d_0=1\) and \(d_i\) is a greatest common divisor of the \(i\times i\) minors of \(A\).
Two \(m\times n\) matrices over \(R\) are equivalent if and only if they have the same Smith normal form.
The computation of \(\SNF(A)\) by row and column operations is in [DuFo04].
(Classification of finitely generated modules)
If \(M\cong R^n/\im(A)\) for a matrix \(A\in\Mat(n\times m;R)\), then \[ M\cong R^{n-r}\oplus\bigoplus_{i=1}^r R/\gens{a_i}, \] where \(a_1,\ldots,a_r\) are the nonzero diagonal entries of \(\SNF(A)\); the nonunit \(a_i\) are the invariant factors of the torsion submodule of \(M\). This is the computation on Classify this module. For \(R = \ZZ\) it classifies finitely generated abelian groups, and for \(R=k[x]\) applied to \(xI-A\) it gives the invariant factors of the rational canonical form of a square matrix \(A\) over a field \(k\).