Given an algebra $A$ with a right $A$-module $M$ with $End\_A(M) \cong A$.
Then we can view $M$ as a natural $A$-bimodule. When is $M$ as a bimodule indecomposable and what is its endomorphism ring as a bimodule?


In fact in my examples $M$ was always indecomposable and the endomorphism ring was isomorphic to the center of $A$ (but I did not calculate many examples, since my computer can not do that).



Endomorphism ring of bimodules
The endomorphism ring of such a bimodule is indeed always the center: If $f$ is a bimodule-endomorphism of $M$, then in particular $f\in End({\_A M})$ so that by assumption $f(m)=ma$ for some $a\in A$. If this is also a right-module homomorphism, then $\forall b,m: mab = f(m)b = f(mb) = mba$. Because the right action is faithful, this means $\forall b: ab=ba$.


This also shows that $M$ is indecomposable iff $Z(A)$ has no non-trivial idempotents iff $A$ is indecomposable as an algebra.


