As a special case:
Suppose $A$ is a finite-dimensional algebra over a finite field $K$, and $S$ is a (finite-dimensional) simple $A$-module. Let $D := \mathrm{End}_A(S)$. Show that then $D$ must be a field.
I have completely no idea about proving the endomorphism ring commutative, for example where I could apply the condition that $K$ is finite?(I know $A, S$ and endomorphism ring must be finite) I want to start with simpleness condition proving the kernel of the commutator is non-trivial, so I'm hoping the finiteness condition would provide me with such a vector. But I find maybe I also need the condition that the endomorphism is commutative with elements in A, but A is too general to provide any useful information helping locate such vector. So can someone give me a little hint about this?
PS. This is an exercise after Schur’s Lemma. Instead of referencing Wedderburn's lemma, I thought there might be a more direct approach.