This is a more structured reformulation of this question.
Let $k$ be a field and $A$ be a commutative $k$-algebra, say $A=k[x_1,\dotsc,x_n]$, and $M$ be an indecomposable finite dimensional (ungraded or $\mathbf{Z}^n$-graded) $A$-module. Denote by $E=\operatorname{End}_A(M)$ the ring of (homogeneous) endomorphisms of $M$. I am looking for examples (or proofs that this cannot occur) for the following situations:
$$\begin{array}{l|cc} & \text{$M$ is an $A$-module} & \text{$M$ is an $\mathbf{Z}^n$-graded $A$-module}\\ \hline k\subsetneq E & \begin{aligned}A&=k[x]\\ M&=k[x]/(x^2+1)\\E&=k[x]/(x^2+1)=k[i]\end{aligned} & * \\ k\subsetneq E/J(E) & ” & ? \\ \text{$E$ is not commutative} & ? & ?\\ \text{$E/J(E)$ is not commutative} & ? & ?\\ J(E)\subsetneq (a\in E\mid \text{$a$ nilpotent}) & ? & ? \end{array}$$
What I know so far:
- an example for * has already been given in here.
- The statement in the last row cannot occur if $E$ is commutative.
Question: What can be said about the other slots?