I am reading Chapter IV, section 2 in Assem's book Elements of the Representation Theory of Associative Algebras and I am stuck at a claim:
Let $A$ be a finite dimensional algebra over $\mathbb{C}$. Then $Hom_{A}(eA, A) \cong Ae$ where $e$ is a primitive idempotent.... (*) This is needed to prove that $Hom_{A}(P, A)$ is a projective left $A$ module if $P$ is a finite dimensional projective right $A$ module.
What I get is $Hom_{A}(eA, A) \cong A$, since I only need to assign an image to $e$ to determine a homormorphism from $eA$ to $A$. So where am I wrong? HOw to get $Ae$?