# Maximal ideals in $\operatorname{End}_A(M)$

$$\newcommand\End{\operatorname{End}}$$Let $$A=k[x_1, \dotsc, x_n]$$ and $$M$$ be a graded $$A$$-module. Let $$E=\End_*(M)$$ be the graded endomorphism ring of $$M$$ (i.e., $$E_i$$ consists of all degree-$$i$$-endomorphisms of $$M$$). What can be said about the Jacobson radical $$J(E)$$ of $$E$$ in terms of the module $$M$$? For instance, if $$M$$ is finite dimensional, all indeterminates $$x_i$$ are centrally nilpotent in $$E$$, and hence are contained in $$J(E)$$. But how to treat this in general?

Specific questions are:

• If $$M$$ is finite dimensional and indecomposable, is it true that $$J(E)=(x_1, \dotsc, x_n)$$?
• Is there a correspondence between maximal ideals of $$E$$ and certain properties of $$M$$?
• How to think of elements of $$J(E)$$ which are not nilpotent?
• what are nilpotent elements of $$E$$ not contained in $$J(E)$$?