Finitely Generated Modules over PIDs

The structure theorem for finitely generated modules over a PID is one of the most powerful results in algebra, simultaneously generalizing the classification of finite abelian groups and the Jordan canonical form.

The Structure Theorem

Theorem9.1Structure theorem for modules over a PID

Let $R$ be a PID and $M$ a finitely generated $R$ -module. Then:

$M \cong R^r \oplus R/(d_1) \oplus R/(d_2) \oplus \cdots \oplus R/(d_s),$

where $r \geq 0$ and $d_1, d_2, \ldots, d_s$ are nonzero non-unit elements of $R$ with $d_1 \mid d_2 \mid \cdots \mid d_s$ . The integer $r$ (the rank) and the ideals $(d_1), \ldots, (d_s)$ are uniquely determined. The $d_i$ are called the invariant factors.

Alternatively, the torsion part decomposes as $R/(p_1^{e_1}) \oplus \cdots \oplus R/(p_t^{e_t})$ where the $p_i$ are primes (the elementary divisors).

Applications

ExampleInstances of the structure theorem

$R = \mathbb{Z}$ : Finitely generated abelian groups: $M \cong \mathbb{Z}^r \oplus \mathbb{Z}/d_1 \oplus \cdots \oplus \mathbb{Z}/d_s$ . Example: $M \cong \mathbb{Z}^2 \oplus \mathbb{Z}/6 \oplus \mathbb{Z}/30$ .
$R = k[x]$ : $M$ is a $k[x]$ -module, i.e., a finite-dimensional $k$ -vector space $V$ with a linear operator $T$ ( $x$ acts as $T$ ). Then $V \cong k[x]/(f_1) \oplus \cdots \oplus k[x]/(f_s)$ with $f_1 \mid \cdots \mid f_s$ . The $f_i$ are the invariant factors of $T$ , and $f_s$ is the minimal polynomial. The characteristic polynomial is $\prod f_i$ .
Jordan form: Over $k = \overline{k}$ , elementary divisors are $(x - \lambda_i)^{e_i}$ , giving Jordan blocks.
Rational canonical form: The invariant factors give the companion matrix decomposition.

Proof Idea

RemarkProof via Smith normal form

The proof reduces to linear algebra over $R$ : any $m \times n$ matrix over a PID can be reduced (by row and column operations) to Smith normal form: a diagonal matrix with entries $d_1, \ldots, d_k, 0, \ldots, 0$ where $d_1 \mid d_2 \mid \cdots \mid d_k$ . This gives the structure theorem by expressing $M$ as a quotient of a free module $R^n$ by the relations encoded in the matrix.