Endomorphisms of a semisimple module Is there an easy way to see the following: 
Given a $k$-algebra $A$, with $k$ a field, and a finite dimensional semisimple $A$-module $M$. Look at the natural map $A \to \mathrm{End}_k(M)$ that sends an $a \in A$ to 
$$
M \to M: m \mapsto a \cdot m.
$$
Then the image of $A$ is a finite-dimensional semisimple algebra. 

Here's another slightly different approach. Let $B$ be the image algebra, which is a finite dimensional algebra over $k.$ Note that $M$ is a $B$-module and its $B$-submodules are precisely the $A$-submodules of $M.$ Hence $M$ is also a semisimple $B$-module, and the Jacobson radical $J(B)$ annihilates $M.$ Thus $J(B) = \{0 \} $ and $B$ is semisimple.
($B$ is a genuine subalgebra of ${\rm End}_{k}(M),$ so is faithfully embedded there, and the elements of $J(B)$ are annihilating $M$, so only the zero endomorphism can be in $J(B) ).$
