Let $M$ be a non-zero module, let $N$ be a proper submodule, and let $x \in M/N$. Prove that:

Question

(1) $M$ has a submodule $K$ maximal with respect to $N \leq K$ and $x \notin K$.

(2) if $M = Rx + N$, then $M$ has a maximal submodule $K$ with $N \leq K$ and $x \notin K$.

Basically, I don't even know how to start.

Answer 1

You clearly have to assume $x\notin N$.

Consider the family $\mathscr{F}$ of submodules of $M$ containing $N$ and not containing $x$, ordered by inclusion. Such a family satisfies the hypotheses of Zorn's lemma, so it has a maximal element.

Point 2 is obviously a particular case (again, provided $x\notin N$).