Classify finitely-generated modules over \(\ZZ\), over PIDs, and over Dedekind rings.
Solution
Let all modules be finitely generated.
1. Over \(\ZZ\), every module is a finitely generated abelian group and has a unique invariant-factor decomposition\[ M\cong \ZZ^r\oplus \ZZ/d_1\ZZ\oplus\cdots\oplus\ZZ/d_t\ZZ, \qquad 1<d_1\mid d_2\mid\cdots\mid d_t. \]Equivalently, its torsion part has a unique elementary-divisor decomposition into cyclic groups of prime-power order.
This is the structure theorem for finitely generated modules over the PID \(\ZZ\).
2. More generally, if \(R\) is a PID, then\[ M\cong R^r\oplus R/(a_1)\oplus\cdots\oplus R/(a_t), \qquad 0\ne a_1\mid a_2\mid\cdots\mid a_t, \]with the nonunit invariant factors unique up to associates.
This is the structure theorem for finitely generated modules over a PID. The free rank \(r\) and the invariant factors determine the isomorphism class. Factoring the \(a_i\) into prime powers gives the equivalent elementary-divisor form.
3. Let \(R\) be a Dedekind domain. Then every finitely generated module decomposes as\[ M\cong P\oplus T, \]where \(T\) is the torsion submodule and \(P\) is finitely generated projective.
Over a Dedekind domain, finitely generated torsion-free modules are projective. The quotient \(M/T\) is torsion-free, hence projective, so the exact sequence \[ 0\to T\to M\to M/T\to0 \] splits. Put \(P=M/T\).
4. If \(P\) has positive rank \(r\), then\[ P\cong R^{r-1}\oplus I \]for a nonzero fractional ideal \(I\), and the ideal class \([I]\in\operatorname{Cl}(R)\) is uniquely determined by \(P\).
This is the Steinitz classification of finitely generated projective modules over a Dedekind domain. Two modules \(R^{r-1}\oplus I\) and \(R^{r-1}\oplus J\) are isomorphic iff \(I\) and \(J\) represent the same ideal class.
5. The torsion part has a unique primary decomposition\[ T\cong \bigoplus_{\mathfrak p} \bigoplus_j R/\mathfrak p^{e_{\mathfrak p,j}}, \]where only finitely many nonzero prime ideals \(\mathfrak p\) occur and, for each \(\mathfrak p\), the exponents may be ordered increasingly.
A finitely generated torsion module over a Dedekind domain has finite support. Localizing at a nonzero prime \(\mathfrak p\) gives a finite-length module over the DVR \(R_{\mathfrak p}\), hence a direct sum of cyclic modules \(R_{\mathfrak p}/\mathfrak p^{e}R_{\mathfrak p}\). Gluing the primary pieces recovers the displayed global decomposition, with uniqueness inherited from the local DVR classifications.
Thus the PID theorem is recovered from the Dedekind classification exactly when the class group is trivial: then every ideal \(I\) is principal and every projective module is free.