In Farb's Noncommutative Algebra, the following exercise is given.
Theorem 3.22 Let $R$ be a finite dimensional central simple $k$-algebra. Every $k$-linear derivation on $R$ is inner.
I solved the exercise, but I noticed that the solution didn't use "the full power" of Noether-Skolem theorem. This made me wonder if the following was true:
If $R$ is a finite dimensional central simple $k$-algebra and $S \subseteq R$ is a simple $k$-subalgebra, then every $k$-linear derivation $d: S \to R$ is inner.
I didn't see an immediate way to generalize the proof of the first statement to the second one, since the second statement requires us to work with the centralizer of $S$ instead of the center of $R$ (which is a field). Any help or hints are appreciated.