Modules over a PID

Key Theorems

Theorem (Submodules of Free Modules over a PID).
Let $D$ be PID, and let $M \subseteq D^n$ be a submodule. Then:

  1. $M$ is a free $D$–module.
  2. There exist elements $x_1, \ldots, x_n \in D^n$ and nonzero elements $a_1, \ldots, a_m \in D$ such that:
    1. $a_1 \mid a_2 \mid \cdots \mid a_m$ (divisibility chain).
    2. $D^n = \bigoplus_{i=1}^n D x_i$ (i.e., ${x_1, \ldots, x_n}$ is a basis of $D^n$).
    3. $M = \bigoplus_{i=1}^m D (a_i x_i)$.

Corollary (Structure Theorem for Finitely Generated Modules over a PID).
Let $D$ be a PID, and let $M$ be a finitely generated $D$–module. Then there exist nonzero elements $d_1, \ldots, d_m \in D$ with

\[d_1 \mid d_2 \mid \cdots \mid d_m\]

such that

\[M \;\cong\; D^r \;\oplus\; \bigoplus_{i=1}^m D / \langle d_i \rangle.\]

Here:

  1. $r = \operatorname{rank}(M)$ is uniquely determined.
  2. The ideals $\langle d_i \rangle$ are uniquely determined (up to multiplication by units).
  3. The minimal number of generators of $M$ is $r + m$.

  4. The torsion submodule of $M$ is

    \[\operatorname{Tor}(M) \;=\; \{m\in M: \exists \,d\in D \text{ s.t }dm = 0\} \;=\; \bigoplus_{i=1}^m D / \langle d_i \rangle.\]

Remarks

  • This decomposition splits $M$ into a free part ($D^r$) and a torsion part (finite direct sum of cyclic modules).
  • For $D = \mathbb{Z}$, the theorem is the Fundamental Theorem of Finitely Generated Abelian Groups.
  • The elements $d_1 \mid d_2 \mid \cdots \mid d_m$ are called the invariant factors of $M$.