I am reading this paper and have the following questions.
Let $A$ be a finite-dimensional algebra over a fixed field $k$.
Let the finitely-generated $A$-module $M$ be a generator–cogenerator for $A$, which means that all projective indecomposable $A$-modules and all injective indecomposable $A$-modules occur as direct summands of $M$.
On page 3 in the paper it says
"The identity of $End_A(M)$ is the sum of the “identity maps” on the indecomposable direct summands of $M$. Hence we have primitive idempotents of $End_A(M)$ corresponding to the summands of $M$. For any indecomposable summand $T$ of $M$ we denote the corresponding simple $End_A(M)$-module by $E_T$".
My questions are:
- How can I prove this correspondence and
- Why is $E_T$ simple?.
I would be grateful for references concerning literature or every other kinds of hints.
Thank you very much.