What Is the Structure of Modules Over Polynomial Rings?

[mathmari]

Hey!

Let $R$ be a commutative ring with unit.

We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ( $a\in R,x\in M$ ) is defined as usual if $a\in \mathbb{R}$, and $a\cdot x=(x_1, 0, 0)$ if $a=t, x=(x_1, x_2, x_3)$.

From the structure theorem for finitely generated $R$-module there is a list $p_1, p_2, \dots , p_n$ irreducible of $R$ and a list $k_1, k_2, \dots , k_n$ positive integers such that $$M\cong R/\langle p_1^{k_1}\rangle \oplus \dots \oplus R/\langle p_n^{k_n}\rangle$$

How can we find these lists? (Wondering)

 

[Deveno]

This is similar to your last problem, although it may not appear so.

To whit, $t$ acts upon $v \in \Bbb R^3$ as the linear transformation:

$T(x,y,z) = (x,0,0)$.

Again, $T$ here satisfies the polynomial $t^2 - t$, and this is its minimal polynomial, but its characteristic polynomial is $t^2(t - 1)$.

We can immediately decompose $\Bbb R^3$ into the eigenspace decomposition:

$\Bbb R^3 = E_0 \oplus E_1$ where $E_0 = \text{ker }T$, and $E_1 = \{(a,0,0): a \in \Bbb R\}$.

It is clear that $E_1$ corresponds to $\Bbb R[t]/\langle t-1\rangle$ since

$(t - 1)\cdot (x,y,z) = t\cdot (x,y,z) - (x,y,z) = (x,0,0) - (x,y,z) = (0,-y,-z)$

so that $t - 1$ annihilates $(x,y,z)$ if and only if $y = z = 0$. Since $t - 1$ is irreducible, we have found one of the factors on our list.

Now you know that $E_0$ corresponds to the annihilator of the subspace $W = \{(0,b,c): b,c \in \Bbb R\} \cong \Bbb R^2$.

What possible irreducible factors of $t^2(t-1)$ are there that annihilate this subspace?