(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.
Let $M$ be a non-zero module, let $N$ be a proper submodule, and let $x \in M/N$. Prove that:
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abstract-algebra
modules
1 Answers
3
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 $x\notin N$).
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0By the way, one cannot avoid Zorn's lemma, because a particular case is $M=R$, $N=\{0\}$ and $x=1$. – 2017-02-12