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I asked this question in mathoverflow. But it was closed. So I ask it here.

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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A complete set of centrally primitive idempotents $\{e_1,\ldots, e_n\}$ in $R$ furnishes a decomposition of of $R$ into directly indecomposable rings, namely the indecomposable rings $e_iRe_i$. A ring may have a complete set of centrally primitive idempotents without having a complete set of primitive idempotents.

These idempotents do not connect well with the category of $R$ modules. For instance, in any simple ring (or finite product of simple rings) you always have a complete set of centrally primitive idempotents. Modules for simple rings vary a lot, so you can't say much about their modules.

In contrast, a complete set of primitive idempotents provides a much finer grained picture of the left and right ideals of $R$, and therefore of the category of modules.

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  • $\begingroup$ thank you very much. Usually when we want to study the ordinary quiver of an algebra, we compute a complete set of primitive orthogonal idempotents. Does a complete set of primitive idempotents contain a complete set of primitive orthogonal idempotents? $\endgroup$
    – LJR
    Mar 27, 2015 at 2:03

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