Structure theorem for finitely generated modules over a PID: invariant factors and elementary divisors

Let \(R\) be a PID and let \(M\) be a finitely generated \(R\)-module. Then \[ M\cong R^r\oplus\bigoplus_{i=1}^s R/(d_i), \] where \[ d_1\mid d_2\mid\cdots\mid d_s \] and each \(d_i\) is nonzero and nonunit. The integer \(r\) and the invariant factors \(d_i\), up to multiplication by units, are unique.

To prove existence, choose a surjection \[ R^n\twoheadrightarrow M \] with kernel \(N\). Since \(R\) is a PID, the submodule \(N\le R^n\) is free of finite rank. Choose bases of \(R^n\) and \(N\) and let \(A\) be the matrix of the inclusion \(N\hookrightarrow R^n\). Smith normal form gives invertible matrices \(U,V\) such that \[ UAV=\operatorname{diag}(d_1,\ldots,d_s,0,\ldots,0), \qquad d_1\mid\cdots\mid d_s. \] Taking the quotient of \(R^n\) by the image gives \[ M\cong R^{n-s}\oplus\bigoplus_{i=1}^s R/(d_i). \]

Factor each invariant factor into prime powers. Since distinct prime powers are comaximal, the Chinese remainder theorem gives \[ R/(d_i) \cong \bigoplus_{p^e\parallel d_i} R/(p^e). \] Thus one obtains the elementary-divisor form \[ M\cong R^r\oplus\bigoplus_j R/(p_j^{e_j}), \] where the \(p_j\) are prime elements, not necessarily distinct.

Conversely, grouping the elementary divisors by prime and aligning equal-prime powers in nondecreasing order reconstructs invariant factors satisfying the divisibility chain.

Uniqueness of the invariant factors follows from Smith normal form, equivalently from the determinantal ideals of a presentation matrix; uniqueness of the elementary divisors follows from unique factorization in the PID together with primary decomposition.