An $R$-module $M$ is called semisimple if one of the following condition holds:
1) $M$ is a direct sum of simple* submodules of $M$
2) $M$ is a sum of simple submodules of $M$
3) For any $R$-submodule $N$ of $M$ there exists $N'$ s.t $N \oplus N'=M$
A ring $R$ is called semisimple if it is semisimple as a module over itself.
(*An $R$-module $M$ is simple if there exits no non-trivial proper submodule of $M$ )
The question is there a direct proof using the definition that $M_n(\mathcal k)$ is a semisimple ring.