Let $R = M_n(k)$, where $k$ is a field. Then any R-module that is finite dimensional over K is a direct sum of 
Let $R = M_n(k)$, where $k$ is a field. Then any $R$-module that is finite dimensional over $K$ is a direct sum of isomorphic copies of $V$, where $V = k^n$. 

I was able to show that $R$ has a unique simple module $V$, but I'm stuck on the rest of the exercise. 
Any help is appreciated. 

The full matrix ring over a field is a semisimple ring, so all of its modules are direct sums of simple submodules.
Since there is exactly only one isotype of simple module, the submodules in those decompositions must be copies of the one simple module. It does not even have anything to do with finite generation.
