structure theorem for modules over a PID

The structure theorem is the great classification result that says: over a principal ideal domain, every finitely generated module is, up to isomorphism, just a direct sum of the simplest possible pieces — some free copies of the ring, plus some cyclic torsion blocks. Nothing more exotic can occur. It is the single theorem that simultaneously classifies finitely generated abelian groups and produces the Jordan and rational canonical forms in linear algebra.

Let R be a PID and M a finitely generated R-module. Then M ≅ R^r ⊕ R/(d_1) ⊕ R/(d_2) ⊕ ... ⊕ R/(d_k), where r ≥ 0 is the free rank and d_1 | d_2 | ... | d_k are nonzero nonunits (the invariant factors), and this data is unique. Equivalently, the torsion part decomposes into prime-power cyclic pieces R/(p^a) (the elementary divisors). The torsion submodule and the free rank r are themselves invariants.

The proof rests on the fact that submodules of a free module over a PID are again free, combined with the Smith normal form of the relations matrix: row and column operations reduce any presentation matrix to a diagonal form with the invariant factors on the diagonal. Applied to R = Z it gives the fundamental theorem of finitely generated abelian groups; applied to R = k[x] with M a vector space made into a k[x]-module by a linear operator, it produces the rational and Jordan canonical forms of that operator.