Finding the submodule and the endomorphism

Question

working on some problems with modules and ran into a problem.

Let $M$ be the $\mathbb R[x]$-module $\mathbb R^2$ where $x$ acts via the linear transformation $$\begin{bmatrix}2 & 0\\0 & 3\end{bmatrix}$$ I need to find all $\mathbb R[x]$-submodule of $M$ and $End_{\mathbb R[x]} (M)$.

Could anyone share some tips on how to get started on this problem?

Thanks!

Answer 1

Hint: The $\mathbb R[x]$-submodules of $M$ are the subspaces invariant under the action of $x$, that is, under the given linear transformation.

For the endomorphism part, note that $\phi$ is an $\mathbb R[x]$-endomorphism iff $\phi(xm)=x\phi(m)$ and this means that $\phi$ is a linear transformation that commutes with the given linear transformation.

Answer 2

Suppose that $(u,v)$ is the basis in which is written the matrix of $x$, $Vect(u)$ and $Vect(v)$ are submodule as well as the zero submodule. If $w\neq 0$ is not an eigenvector, $(w,x(w))$ is free so generated $\mathbb{R}^2$. So the only submodules are $Vect(u), Vect(v)$ and the zero submodules.