= Finite-dimensional simple algebra is semisimple
For a nonzero <associative algebra> $A$ of finite <dimension> that is a <simple ring>, choose a <minimal left ideal> $I$. Right multiplication $I\to Ia$ is a <module homomorphism>, so $Ia$ is zero or a <simple module>. The sum $\sum_{a\in A}Ia$ is a nonzero <two-sided ideal>, hence $A$. Finite <dimension> reduces this to a finite sum. The <finite sum of simple submodules> lemma shows that the <left regular module> is a <semisimple module>.
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