Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's indecomposable modules are determined by Jordan blocks $$ J_{\lambda,m}= \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix}.$$ The simple modules are precisely the 1-dimensional modules. For every simple module $M_1$ defined such that the action of $x$ on $M_1$ is scalar multiplication by $\lambda \in K$, we have an infinite chain of submodules $$ M_1 \subset M_2 \subset M_3 \subset \ldots,$$ where the action of $x$ on $M_i$ is a linear transformation by $J_{\lambda, i}$. In particular, such a chain of indecomposables induces a homogeneous tube in the Auslander-Reiten quiver $\Gamma$ of finite-dimensional modules of $K[x]$, and all components of $\Gamma$ are homogeneous tubes with a simple module at the mouth. As a consequence, there is an irreducible monomorphism $M_i \rightarrow M_{i+1}$ and irreducible epimorphism $M_{i+1} \rightarrow M_i$ for all $i$ in the chain above. If
$$ M'_1 \subset M'_2 \subset M'_3 \subset \ldots,$$ is a different chain (so $M_1 \not\cong M'_1$), then $\mathrm{Hom}_{K[x]}(M_i, M'_j)=0$ and $\mathrm{Hom}_{K[x]}(M'_i, M_j)=0$ for any $i$ and $j$.

My questions consist of two parts:

(1) What is known about the finitely-generated (but not necessarily finite-dimensional) representation theory of $K[x]$? Is the category of finitely-generated representations tame (i.e. is it possible to classify the indecomposable representations and/or $\mathrm{Hom}$-spaces? An explanation/summary accompanied by references would be greatly appreciated (if this is feasible, that is).

(2) Is anything at all known about the infinite-dimensional (but not necessarily finitely-generated) representation theory of $K[x]$? My instinct says that this category is likely wild (but a confirmation/rebuttal is welcome). Even so, are there known examples or families of infinitely-generated representations that could be considered interesting, or could have potential applications to other areas of mathematics (such as mathematical physics)?

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    $\begingroup$ There is a large literature on (possibly infinitely generated) abelian groups (=$\mathbf{Z}$-modules), and most of it can immediately be adapted to modules over PIDs. You get plenty of natural classes: artinian, minimax, locally finite dimensional, torsion-free of finite rank... there's a lot more to say, I'm not sure what to start with if you're not more specific. $\endgroup$ – YCor Feb 26 at 16:25
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    $\begingroup$ (1) Yes, it's immediate from the theorem of classifications of modules over a PID. In particular, you get (free)$\oplus$(finite-dimensional) $\endgroup$ – YCor Feb 26 at 16:26
  • $\begingroup$ So does your second comment imply that there is only one indecomposable $K[x]$-module that is infinite-dimensional and finitely-generated -- namely the free module $K[x]$? $\endgroup$ – Iteraf Feb 26 at 17:01
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    $\begingroup$ Yes. It also works for, say, the ring $K[x,x^{-1}]$. Back to $K[x]$, there's no assumption that $K$ is algebraically closed. So the result says that a f.g. indecomposable over $K[x]$ is isomorphic to either $K[x]$ itself, or to $K[x]/P(x)^n$ for some monic irreducible polynomial $P$ of positive degree and $n\ge 1$. $\endgroup$ – YCor Feb 26 at 17:02

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