Let $M$ as a $R$-module and $N\subset M$ a fully invariant submodule of $M$. Why the exists a maximal submodule of $M$ such as $L$ that $L\cap N=\{0\}$.
Definition: We refer $N$ as a fully invariant submodule of $M$ if for every homomorphim $\varphi : M\to M$ we have $\varphi(N)\subset N$.
Zorn's Lemma: define
$$C:=\left\{\,K\le_R M\,\;/\;K\cap N=\{0\}\right\}$$
since $\;\{0\}\in C\;$ we have that $\;C\neq\emptyset\;$ , and it's easy (do it) to see $\;C\;$ is partially ordered by set inclusion and fulfills the conditions on any chain in it , so it has a maximal element...