Uniqueness
While the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between M and its canonical form is not unique, and does not even preserve the direct sum decomposition. This follows because there are non-trivial automorphisms of these modules which do not preserve the summands.
However, one has a canonical torsion submodule T, and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:
Compare composition series in Jordan–Hölder theorem.
For instance, if, and is one basis, then is another basis, and the change of basis matrix does not preserve the summand . However, it does preserve the summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).
Read more about this topic: Structure Theorem For Finitely Generated Modules Over A Principal Ideal Domain
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