Semisimple algebra
= Semisimple algebra
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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 $e=0$.