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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.

  • 0
    @Aaron Maybe yo$u$ want to convert your comment here to an answer?2013-06-14

0 Answers 0