## A module \$N\$ is semisimple \$\Longleftrightarrow\$ \$N\$ has no proper essential submodules

Problem: A module $$N$$ is semisimple $$\Longleftrightarrow$$ $$N$$ has no proper essential submodules.

My attempt: If $$N$$ is semisimple, then every submodule is a direct summand of $$N$$ and so not essential unless equal to $$N$$. Conversely, any $$K \subsetneq N$$ has a complement $$L \subsetneq N$$. Then $$K \bigoplus L \subseteq N$$, so if $$N$$ has no proper essential submodules, $$K$$ is a direct summand of $$N$$.

Please check my proof. Thank all!