The structure theorem over a PID is the algebraic reason finitely generated abelian groups and many canonical forms can be classified. It is a module theorem before it is a classification theorem.
Finitely generated modules over a PID
Theorem. If \(R\) is a PID and \(M\) is a finitely generated \(R\)-module, then \[ M\cong R^r\oplus R/(d_1)\oplus\cdots\oplus R/(d_t), \] where \(d_i\ne0\), \(d_i\) is not a unit, and \(d_1\mid d_2\mid\cdots\mid d_t\). The integer \(r\) and the ideals \((d_i)\) are uniquely determined.
Definition. The elements \(d_1,\ldots,d_t\) are invariant factors. The summand \(R^r\) is the free part, and the quotient summands form the torsion part.
Smith normal form
Theorem. If \(A\) is a matrix over a PID, then elementary row and column operations transform \(A\) into a diagonal matrix \[ \operatorname{diag}(d_1,\ldots,d_t,0, \ldots,0), \qquad d_1\mid d_2\mid\cdots\mid d_t. \] This is the Smith normal form.
Proof. The Smith form algorithm repeatedly chooses a nonzero entry of minimal divisibility, uses row and column operations to make it divide all entries in its row and column, clears those entries, and then continues on the remaining submatrix. The PID hypothesis is what makes the gcd and divisibility steps possible. The diagonal form gives generators with relations \(d_ie_i=0\).
Remark. Smith normal form classifies cokernels of maps \(R^m\to R^n\). The structure theorem follows by presenting a finitely generated module as generators modulo relations.
Finitely generated abelian groups
Corollary. Every finitely generated abelian group is isomorphic to \[ \mathbb Z^r\oplus\mathbb Z/d_1\mathbb Z\oplus\cdots\oplus\mathbb Z/d_t\mathbb Z, \qquad d_1\mid d_2\mid\cdots\mid d_t. \]
Example. The abelian group \(\mathbb Z/12\mathbb Z\oplus\mathbb Z/18\mathbb Z\) can also be expressed through primary components: \[ \mathbb Z/4\mathbb Z\oplus\mathbb Z/3\mathbb Z\oplus \mathbb Z/2\mathbb Z\oplus\mathbb Z/9\mathbb Z. \] Combining coprime factors recovers invariant factors.
Elementary divisors
Theorem. The torsion part can also be decomposed as a direct sum of \(R/(p_i^{a_i})\), where the \(p_i\) are prime elements. These cyclic prime-power summands are the elementary divisors.
Remark. Invariant factors emphasize divisibility chains. Elementary divisors emphasize prime-power primary decomposition. The two forms carry the same information.
Linear operators as modules over F[x]
Definition. If \(T:V\to V\) is a linear operator over \(F\), then \(V\) becomes an \(F[x]\)-module by \(f(x)\cdot v=f(T)v\). The structure theorem over \(F[x]\) classifies \(T\) up to change of basis.
Corollary. Rational canonical form and Jordan form are module decompositions in disguise. The invariant factors correspond to companion blocks; over an algebraically closed field, elementary divisors correspond to Jordan blocks.
A concrete Smith calculation
Calculation. For \[ A=\begin{pmatrix}2&0\\0&6\end{pmatrix} \] over \(\mathbb Z\), the cokernel is \(\mathbb Z/2\mathbb Z\oplus\mathbb Z/6\mathbb Z\). The divisibility \(2\mid6\) already puts the diagonal in invariant factor form.
Smith normal form
Theorem. For a matrix \(A\) over a PID \(R\), there are invertible matrices \(P,Q\) such that \[ PAQ=\operatorname{diag}(d_1,\ldots,d_r,0,\ldots,0), \qquad d_1\mid d_2\mid\cdots\mid d_r. \] The nonzero \(d_i\) are unique up to units.
Proof outline. Elementary row and column operations preserve the presented module. Use Bezout operations to replace nonzero entries until one entry \(d_1\) divides every entry; move it to the upper-left corner and clear its row and column. Repeat on the remaining submatrix. If adjacent diagonal entries fail divisibility, a \(2\times2\) Bezout operation replaces them by their gcd and lcm. Uniqueness follows because the ideal generated by all \(k\times k\) minors is \((d_1\cdots d_k)\), an invariant of equivalence.
Classification of finitely generated modules
A presentation \(R^m\xrightarrow{A}R^n\to M\to0\) and Smith form give \[ M\cong R^{n-r}\oplus R/(d_1)\oplus\cdots\oplus R/(d_r). \] Factoring each \(d_i\) into prime powers yields the elementary-divisor form. For \(R=\mathbb Z\), this classifies finitely generated abelian groups; for \(R=F[t]\), applying it to the module where \(t\) acts as \(T\) yields rational canonical form and, after splitting, Jordan form.
Example. If a presentation matrix has Smith form \(\operatorname{diag}(2,12,0)\), then \[ M\cong\mathbb Z/2\mathbb Z\oplus \mathbb Z/12\mathbb Z\oplus\mathbb Z. \] The free rank and torsion invariants are intrinsic, not artifacts of the chosen generators and relations.