Semisimple module example

Question

I need to find an example of a module over $\mathbb{F}[x]$ which is two dimensional over the field $\mathbb{F}$ and not semisimple. I do not know how to do it.

Thanks

Answer 1

A two dimensional module over $\mathbb{F}$ is just a two dimensional vector space $V$. The action of $x$ on $V$ is like a $2\times 2$ matrix. To say that it is semisimple means that it is either simple (has no nontrivial submodule which must be of dimension 1 over $\mathbb{F}$) or decompose into a sum of two simple modules.

In any way, you have one dimesion vector spaces that are invariant under a matrix multiplication, which is another way of saying eigenvector. So basically, the question is to find a matrix that have exactly one eigenvector.