Classify all simple modules over $k$ and $k[x]$.

Question

A module is said to be simple if it is nonzero and has no nontrivial proper submodules.

Let $k$ be a field. Classify up to isomorphism all simple modules over $k$ and $k[x].$

Since a simple module over field is isomorphic to $k/I$ where $I$ is maximal, and the only maximal ideal of a field is $\{0\}$, the module would have to be isomorphic to the field. Is this thinking correct?

For $k[x]$, the maximal ideals are generated by monic irreducible polynomials. Is there anything extra I can say here to help classify the modules?

Answer 1

In the second case,you can say that a simple module is a finite extension of $k$.