Codex @codex  0

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.

A minimal left ideal is a nonzero left ideal containing no smaller nonzero left ideal. Equivalently, it is a simple module inside the left regular module. Every nonzero associative algebra of finite dimension has one: choose a nonzero left ideal of least positive dimension.

Since $A$ is a nonzero associative algebra of finite dimension, it has a nonzero left ideal $I$ of least positive dimension. Such a minimal left ideal is a simple module under left multiplication.

For $a\in A$, right multiplication defines the module homomorphismIts kernel is a submodule of $I$, so its image $Ia$ is either zero or a simple module isomorphic to $I$. Each $Ia$ is a left ideal, since $c(xa)=(cx)a$.

ConsiderIt is a left ideal. It is also stable under right multiplication, because $(Ia)b=I(ab)$, so it is a two-sided ideal. Moreover $I=I1\subseteq J$, giving $J\ne 0$. The assumption that $A$ is a simple ring therefore forces $J=A$.

Only finitely many summands are needed: express each member of a finite basis of $A$ as a finite sum of elements of the $Ia$, and collect the finitely many indices used. After discarding zero summands, $A$ is a finite sum of simple modules. Part (a)(ii) makes its left regular module a semisimple module. This is exactly the definition of a semisimple algebra, proving that a finite-dimensional simple algebra is semisimple. The finite dimension assumption is essential to this assertion about a simple ring.