Let $M$ be a module over a ring $R$ and let $m \in M$, where $m$ is non-zero.
I'm trying to show:
$(1) -$ There exists at least one submodule $K$ of $M$,with $m \notin K$ such that this $K$ is the maximal submodule(with respect to containment) which doesn't contain $m$.
I'm not sure where to start with this one at all? The only submodule I can think of given the above information is $Rm$, but this obviously contains $m$.
$(2)$ - $(Rm +K)/K$ is a simple submodule of $M/K$
This is a submodule as if we take $x, y \in (Rm +K)/K$, then $x = rm + K$, $y = r'm + K$, then we can check that $x - y \in (Rm +K)/K$ and $tx, t \in R$, is in $(Rm +K)/K$.I don't know how to show it is simple, though?
Hints:
$(1)$ – Use Zorn's lemma.
$(2)$ – A submodule of $(Rm+K)/K$ has the form $L/K$, where $L$ is a submodule of $M$ such that $$ K\subseteq L\subseteq Rm+K. $$
Now consider the element $m$. What can you say about $L$ if it does not contain $m$. What if it contains $m$?