I need some help on this exercise from A Course in Ring Theory by Donald S. Passman
Find all finitely generated graded $K[x]$-modules up to abstract isomorphism. Remember, $K[x]$ is a principal ideal domain.
The result is supposedly similar to the well-known structure theorem in the non-graded case.
So let $M$ be a finitely generated $K[x]$-module with a minimal generating set of homogeneous elements $\alpha_1, \ldots, \alpha_n$ with $d_i$ the degree of $\alpha_i$ (such a set exists because every element of M is a sum of homogeneous elements). We then get a surjective graded homomorphism $$\phi \colon K[x](d_1) \oplus \ldots \oplus K[x](d_n) \to M,\;e_i \mapsto \alpha_i,$$ where $K[x](d_i)$ denotes the graded module $K[x]$ with its grading shifted upwards by $d_i$ and $e_i = (0,\ldots,1,\ldots,0)$. So the homomorphism theorem for graded modules states that $M$ is - as a graded $K[x]$-module - isomorphic to $$\left(\phi \colon K[x](d_1) \oplus \ldots \oplus K[x](d_n)\right)/\ker(\phi).$$ How do I go on now?
Obviously $\ker(\phi)$ is a graded submodule of something like a free-graded $K[x]$-module. Is there maybe an analogue to the elementary divisors theorem for the graded case which lets me express this quotient in a nice way ?
Thanks for helping me out!