Semisimple algebra

A finite-dimensional associative algebra is semisimple when its left regular module is a direct sum of simple modules. Equivalently, every left ideal is a direct summand. In particular, a semisimple algebra has no nonzero square-zero left ideal: if $L^2=0$ and a module projection $A\to L$ exists, its value $e$ at $1$ satisfies $e\in L$ and $e^2=e$, forcing $e=0$.

If the left regular module of a unital ring is semisimple, then every left module is semisimple. Every module is a quotient of a direct sum of copies of the regular module, hence a quotient of a sum of simple modules. Their simple or zero images span the quotient. A maximal independent direct sum of simple submodules must span: otherwise one of these simple images has zero intersection with it and can be added. Thus the quotient is a direct sum of simples. For an infinite sum the maximality argument uses Zorn's lemma.

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.