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).