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!
2026-05-06 06:37:45.1778049465
Quotient module and submodule.
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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 $\mathbb{Z}$.