LEC ALG4 11

Proposition 407.1(Proposition).

Let $R$ be a PID and $M$ be a finitely generated torsion $R$-module. There exists unique nonzero elements $q_1,\dots ,q_s\in R$ such that $q_1|q_2|\dots |q_s$ and

$$M\cong \frac{R}{⟨q_1⟩}\oplus \cdots \oplus \frac{R}{⟨q_s⟩}.$$

5d2c36

[!Proof]-
What you’d expect; use Thm 111.2. Look up uniqueness though.

Combining the concluding remarks of LEC ALG4 5 and Prp 1, we finally have a classification theorem for finitely generated modules over a PID.

Theorem 407.2(Theorem).

Let $R$ be a PID and $M$ be a finitely generated $R$-module. There exists unique nonzero elements $q_1,\dots ,q_s\in R$ such that $q_1|q_2|\dots |q_s$ and $n⩾0$ such that

$$M\cong \frac{R}{⟨q_1⟩}\oplus \cdots \oplus \frac{R}{⟨q_s⟩}\oplus R^n.$$

1be818

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Rational canonical form

Definition 407.3(Definition).

Let $k$ be a field. The companion matrix of a monic polynomial $x^n+{\sum }_{i=0}^{n-1}{a}_i{x}^i\in k[x]$ is

$$\left[\begin{array}{ccccc}0& 0& 0& \dots & -a_0\\ 1& 0& 0& \dots & -a_1\\ 0& 1& 0& \dots & -a_2\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & -a_{n-1}\end{array}\right],$$

i.e, the matrix of a linear transformation $T$ in the basis $\{\overline{1},\dots ,{\overline{x}}^{n-1}\}$ (…)

[!Definition] Rational canonical form
Let $k$ be field, $V$ a $k$-vector space. Let $T:V\to V$ be $k$-linear. Let $q_1(x),\dots ,q_s(x)\in k[x]$ be such that $q_1|\dots |q_s$ and

$$V\cong \frac{k[x]}{⟨q_1⟩}\oplus \cdots \oplus \frac{k[x]}{⟨q_s⟩}$$

as a $k[x]$-module. The rational canonical form of $T$ is a block diagonal matrix whose blocks are companion matrices of the summands above.