Semisimple algebra
= Semisimple algebra
{wiki=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 $e=0$.