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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.

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Sure: just show $k^n$ is a simple right $R$ module where the module action is just matrix multiplication.

Then notice the subset of matrices which are zero outside of row i, call it $R_i$, are isomorphic to that simple module. Then notice $R\cong \oplus R_i$.

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