Structure theorem for finitely generated modules over a PID

Let \(R\) be a PID and let \(M\) be a finitely generated \(R\)-module. Then there are unique integers \(r\ge0\) and nonzero nonunits \[ d_1\mid d_2\mid\cdots\mid d_s \] (up to associates) such that \[ M\cong R^r\oplus R/(d_1)\oplus\cdots\oplus R/(d_s). \] Equivalently, the torsion part may be decomposed into primary cyclic summands.

To prove this, choose generators of \(M\). There is a surjection \[ R^n\twoheadrightarrow M \] with kernel \(K\). Since submodules of finite free modules over a PID are free, choose a basis of \(K\) and obtain a presentation \[ R^m\xrightarrow{A}R^n\longrightarrow M\longrightarrow0. \] Over a PID, elementary row and column operations reduce \(A\) to Smith normal form \[ UAV=\operatorname{diag}(d_1,\ldots,d_s,0,\ldots,0), \qquad d_1\mid\cdots\mid d_s. \] The invertible matrices \(U,V\) merely change bases in the free modules, so they do not change the cokernel. Therefore \[ M\cong\operatorname{coker}(A) \cong R^{n-s}\oplus\bigoplus_{i=1}^sR/(d_i). \] Uniqueness follows from the determinantal divisors, or equivalently from uniqueness of Smith normal form up to associates.

For finitely generated abelian groups, take \[ R=\ZZ, \] and regard the abelian group \(A\) as a \(\ZZ\)-module by repeated addition. The theorem becomes \[ A\cong\ZZ^r\oplus\ZZ/d_1\ZZ\oplus\cdots\oplus\ZZ/d_s\ZZ, \qquad d_1\mid\cdots\mid d_s. \]