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!