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Let $M$ be a module over a ring $R$. My question is:

Does there exists a submodule $N$ such that $M/N$ is isomorphic to $Ra$ for some $a\in M\setminus N$, that is, $M/N$ is 1-'dimensional'?

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No. Take $R=\mathbb Z$ and $M=\mathbb Q$. Then every quotient module $M/N$ is divisible, so it can't be cyclic.

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    $\begingroup$ Note that the property holds for finitely generated modules. $\endgroup$ – user26857 Mar 30 '15 at 13:19
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    $\begingroup$ @gopal Look at math.stackexchange.com/questions/110845/… . If $N$ is a maximal submodule of $M$, then $M/N$ is simple, hence is generated by one element. Finitely generated modules have maximal submodules. $\endgroup$ – Crostul Mar 30 '15 at 13:19

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