Uniqueness of invariant factors and elementary divisors over a PID

An isomorphism carries torsion elements to torsion elements, so it induces an isomorphism of torsion submodules and of the quotients by torsion. The latter quotients are finite free, and invariant basis number recovers their common free rank.

For every associate class of irreducible $p$ and every $k\ge 1$, an isomorphism preserves $M[p^k]$, multiplication by $p^{k-1}$, and the resulting $R\mathrm{/}(p)$-dimension. By [L1], it therefore preserves every elementary-divisor multiplicity.

Aligning the unique prime-power columns as in the primary-decomposition theorem recovers one divisibility chain of invariant factors up to associates. Hence invariant factors are unique as well. Empty torsion data, pure torsion, pure free, and the zero module are included.

Conversely, equal free ranks and equal invariant-factor or elementary-divisor data give termwise isomorphisms between the corresponding direct-sum decompositions; their direct sum constructs a module isomorphism. This proves both directions of the classification.