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Invariant-factor decomposition of a finitely generated module over a PID

Statement

Every finitely generated PID module is a finite free module direct-summed with cyclic torsion quotients. Precisely, if $M$ is finitely generated over a PID $R$ , then

$M ≅ R^{s} \oplus R / (a_{1}) \oplus \dots \oplus R / (a_{t}),$

where each $a_{i}$ is a nonzero nonunit and $a_{1} ∣ \dots ∣ a_{t}$ . Every finitely generated PID module has an invariant-factor decomposition. This assertion is existence; uniqueness is proved separately.

Facts & Assumptions

Given: A finitely generated module $M$ , quotient modules (Quotient module $M / N$ with scalar multiplication on additive cosets), and the first isomorphism theorem for modules (First isomorphism theorem for modules: $M / \ker f ≅ im f$ ).

[L1]

For a submodule $N$ of a free PID module $M$ of finite rank $n$ , there are a basis $e_{1}, \dots, e_{n}$ of $M$ and nonzero elements $a_{1} ∣ \dots ∣ a_{r}$ with $r \leq n$ such that $a_{1} e_{1}, \dots, a_{r} e_{r}$ is a basis of $N$ (Simultaneous bases for a submodule of a finite free module over a PID).

Proof

technique · constructive

1.1

Choose generators $m_{1}, \dots, m_{n}$ of $M$ and define the surjection $π : R^{n} \to M$ by sending the standard basis to them; put $N = \ker π$ . For the zero module one may take $n = 0$ .

givenconstruct

2.1

Apply [L1] to $N \leq R^{n}$ . There is a basis $e_{1}, \dots, e_{n}$ of $R^{n}$ and a basis $a_{1} e_{1}, \dots, a_{r} e_{r}$ of $N$ , with nonzero $a_{1} ∣ \dots ∣ a_{r}$ .

step 1.1L1

3.1

The first isomorphism theorem and coordinatewise quotient give $M ≅ R^{n} / N ≅ R / (a_{1}) \oplus \dots \oplus R / (a_{r}) \oplus R^{n - r}$ . Any unit $a_{i}$ contributes the zero quotient and is removed; the remaining nonunit factors preserve the divisibility chain. This constructs the stated decomposition, including purely free, purely torsion, cyclic, and empty cases.

step 2.1givenalgebradischarge-construct∎

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, Modules over a PID, Theorem 4.1 (standard reference, not scraped)
M. Brussel, Finitely Generated Modules over a PID, Section 3 (standard reference, not scraped)