Finding essential ideal in a ring $R$

Question

Let $R$ be a ring and let $L$ be a left ideal of $R$. The left ideal $L$ is said to be essential if $L \cap S \neq \lbrace 0 \rbrace$, for any non-zero left ideal $S$ of $R$.

Now let $L$ be a random left ideal of $R$. I have to show that there exists a left ideal $L'$ of $R$ such that $L \cap L' =\lbrace 0 \rbrace$ and $L \oplus L'$ is an essential ideal of $R$.

Any tips?

Answer 1

then there are two options: $S∩L≠\{0\}$ (in this case $L⊕L′∩S≠\{0\}$) and $S∩L=\{0\}$. In this case we have that $S∈P$. How do I continue from here?

You're basically right on top of the answer. You've already chosen $L'$ to be maximal (in your comment). If $L\oplus L'\cap S=\{0\}$, then $L\oplus L'\oplus S$ is direct as well.

So $L'\oplus S\in P$, not merely $S$.

Now do you see the contradiction?