structure theorem for f.g. modules over a PID

This is the master theorem of the whole subject. It says that any finitely generated module M over a PID R can be broken into the simplest possible pieces: a free part (some copies of R itself) plus a finite sum of cyclic torsion modules, and this decomposition is essentially unique. Once you have it, both major canonical forms are corollaries, not new theorems.

There are two equivalent forms of the conclusion. Invariant-factor form: M is isomorphic to R^r plus R/(d1) plus R/(d2) plus ... plus R/(dk) with d1 | d2 | ... | dk. Elementary-divisor form: M is isomorphic to R^r plus a sum of pieces R/(p^e), one for each prime power. The integer r is the free rank; the d_i are the invariant factors; the p^e are the elementary divisors.

Specialize R to F[x] and M to a finite-dimensional V viewed as an F[x]-module via T. Then V is a torsion module (r = 0), the cyclic pieces R/(f_i) are exactly cyclic subspaces with companion-matrix action, and the two forms become the rational canonical form (invariant factors) and the Jordan form (elementary divisors, over a splitting field). It is genuinely one theorem wearing two hats.

Two honest caveats. The free rank r is unique, and the invariant factors are unique, but the splitting into cyclic summands is only unique up to isomorphism — the actual submodules you pick are not canonical. And the result needs finite generation: drop it and infinite-rank or non-decomposable modules appear.