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If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$.

If we obtain a complete set of primitive central idempotents of $A$, what modules can be obtain?

I found that some papers compute a complete set of primitive orthogonal idempotents of an algebra while some other papers compute primitive central idempotents. What are the differences between primitive central idempotents and primitive orthogonal idempotents? Thank you very much.

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    $\begingroup$ I think what the OP wants is that a complete set of orthogonal primitive idempotents for the center of A gives a decomposition of A into indecomposable two-sided ideals (i.e. gives a decomposition of A into indecomposable A-A bimodules). $\endgroup$
    – Benjamin Steinberg
    Commented Mar 25, 2015 at 14:30
  • $\begingroup$ @BenjaminSteinberg, thank you very much. I think that if we want to compute the ordinary quiver of an algebra $A$, we need to compute a complete set of primitive orthogonal idempotents of $A$ (but not a complete set of primitive central idempotents of $A$). Is this true? $\endgroup$
    – Jianrong Li
    Commented Mar 28, 2015 at 2:26
  • $\begingroup$ @BenjaminSteinberg, thank you very much. $\endgroup$
    – Jianrong Li
    Commented Mar 28, 2015 at 2:38

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