I have the following question:


Suppose, I have a finite dimensional $k$-Algebra $A$ over an arbitrary field $k$ and a finite dimensional module $M$ that is a generator-cogenerator of mod-$A$.



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> I'm searching for general criteria on $M$ that ensure that the Algebra $B:=End\_A (M)$ is elementary and basic.
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> Moreover, if $B$ isn't elementary, is there a good candidate for a finite field extension $L$ of $k$ such that $B$ is elementary over $L$?
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Thank you very much.



Criteria for a finite-dimensional $k$-Algebra to be basic and elementary
Assume $M$ is the direct sum of the indecomposable modules $M\_i$. Then $B$ is basic iff $M$ is basic, meaning that $M\_i$ is not isomorphic to $M\_j$.
The simple modules then are $End(M\_i)/Rad(End(M\_i))$ , viewing this as a module via projections. Thus the algebra is basic and additionally elementary iff all $End(M\_i)/Rad(End(M\_i))$ are one-dimensional.
How to find such a field extension is for example described in the representation theory book by Zimmermann.


