Tensor Product of irreducible modules

Question

Let $A$ be a $\mathbb C$ algebra. Let $S$ be an irreducible $A$ module?

Then what $ S \otimes_A Hom_A(S,S)$? Is it equal to S? I know that $S \otimes_{\mathbb C} Hom_A(S,S)$ is isomorphic to $S$ as a $\mathbb C$ vector space but what is $ S \otimes_A Hom_A(S,S)$?

Answer 1

The expression $S \otimes_A Hom_A(S,S)$ doesn't make sense, because if $S$ is an irreducible right $A$-module, there is usually not any natural left $A$-module structure on $Hom_A(S,S)$. For instance, if $A=M_n(\mathbb{C})$ for some $n>1$ and $S=\mathbb{C}^n$, then $Hom_A(S,S)=\mathbb{C}$ cannot be made into an $A$-module (in any way compatible with the $\mathbb{C}$-vector space structure).