I am trying to prove the following from Peter Schneider's book : Assuming $A$ is complete and $A/Jac(A)$ is left artinian, for a finitely generated projective indecomposable $A-$module $M$, the quotient $M/Jac(A)M$ is simple. (This generalize the assumption that $A$ is a finite dimensional algebra).
The part $Jac(A)M = Rad(M)$ follows from $M$ being projective and existence of a maximal submodule in $M$ follows from Nakayama's lemma. I am trying to prove $M$ has a unique maximal ideal. Any help?
Here it goes : M/J(A)M has a natural A/J(A)-structure which carries the finite generation property from A-module M. Since A/J(A) is left artinian, using Akizuki-Hopkins-Levinski it is left noetherian and hence M/J(A)M is of finite length. Now M is projective imply Rad(M)=J(A)P. Hence using correspondence theorem of maximal ideals Rad(M/J(A)M)=0. This implies M/J(A)M is semisimple (see Prop1.2.1(iii) ref above) with finitely many simple copies each isomorphic to A/J(A)t_i. Now using A complete lift these t_i to a mutually orthogonal system to M which is a direct sum. But M is indecomposable.