I'm trying to solve a question which asks me to consider the matrix $A$ with
$$A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & -1 \end{bmatrix} $$
What are the submodules of $M$, where $M$ is the $\mathbb{R}[x]$ module defined by $A$ (where for $v \in \mathbb{R}^3$, we have $xv \mapsto Av$. Hence $x^2v\mapsto v$, $x^3v \mapsto Av$ etc.)?
I know the submodules should just be the $A$ invariant subspaces of $\mathbb{R}^3$, but I'm struggling to see if there's a nice way to represent the possible such subspaces. Your help would be appreciated.
As this went unanswered some days I rewrote my comment into an answer.
Actually the problem of finding the invariant subspaces of a given matrix is not an easy task. In this case we are blessed with a diagonal matrix. As it is diagonalizable it decomposes $\Bbb R^3$ as a sum of eigenspaces. That is $\Bbb R^3 = E_1 \oplus E_{-1}$, where $E_i$ are the respective eigenspaces. The eigenspace $E_1=\{ (x_1,x_2,0): x_1, x_2 \in \Bbb R \}$ has dimension two and $E_{-1}=\{ (0,0,x_3): x_3 \in \Bbb R \}$ has dimension one. Clearly the eigenspaces are $A$-invariant subspaces. But what is true still is that the sum of $A$-invariant subspaces is an $A$-invariant subspace. Finally we see that every subspace of the eigenspace $E_1$ is $A$-invariant as the matrix acts as the identity on this eigenspace.
Let $W \subset E_1$ of dimension one. This means all the $A$-invariant subspaces are of the following type. $E_1, E_{-1}, W \oplus E_{-1}$.