I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference.
1) I know that $A$ being self-injective implies that every projective module is injective, but $M$ is only a faithful module...does $M$ have to have the form $Ae$? or does this have something to do with Proposition 2.4 (Morita equivalence of some endomorphism rings)?
2) $A$ can be viewed as a submodule of $M^r$, but why is $A$ in $\mathfrak{add}\ M$ ?
3) How does $A=\text{End}(M_B)$ follow ?
Thanks for the help!