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Set $M$ as an $R$-module, where $R$ is a PID, then if any quotient module of M can be seen as a submodule of M? It seems not always true, and could you give me a counterexample? Thank you!

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    I gave the counterexample in my answer to your previous question http://math.stackexchange.com/questions/1334863/proof-about-finitely-generated-torsion-free-r-module-m-is-free-where-r-is-a-pid – egreg Jun 23 '15 at 10:08

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Did you mean to ask if any quotient module can be seen as submodule?

Not in general: $\mathbb{Z}$ has $\mathbb{Z}/(2)$ as a $Z$-module (abelian group) quotient, but the latter is not a subgroup of $\mathbb{Z}$.

Elle Najt
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