How can i show that a Left-Module M over a finte dimensional simple K-algebra A, that has finite dimension over K, is isomorphic to a direct sum of an minimal left ideal in A and the number of summands is well defined?
My idea is: Because of the finte dimension i know that A has a minimal left ideal I. So i can define the map $I^{n}\to M,(x_1,\dots,x_n) \to m_1x_1+\dots m_nx_n$. I choose n to be minimal so the map is injectiv. But is this map surjectiv? I know that M=AM=IAM=IM. But for this map i need a finte set of generators. I know that there is a finite K-Basis in M but is this an I-Basis?
The natural map $\rho\colon A \rightarrow {\rm End}_K(I)$ defining the action of $A$ on $I$ is injective, since the kernel is a proper two-sided ideal and $A$ is simple. But the map is also surjective because of Burnside's Lemma. Hence $A\cong I^{\oplus n}$.
Reference: Lemma 6 and Theorem 7 here.