The following I regard is the module version of the correspondence theorem
Let $M$ be an $R$ module and $N \leq M$ be an $R$ - submodule of $M$. There exists a bijection $\vartheta:$ submodules of $M / N \longrightarrow$ submodules of $M$ which contain $N$
The lecturer stated this saying that the proof is similar to the rings and groups cases. However, I do not see the map explicitly.
There has been a similar question posted here but it is about the interpretation not the explicit proof.
My question: Can someone give me the exploit map $\vartheta$ that proves the correspondence theorem for modules?
Let $q : M \to M / N$ be the quotient map, sending $m \mapsto [m]_{\text{mod } N}$.
If $S$ is a submodule of $M/N$, then $q^{-1}(S) = \{ m \in M : q(m) \in S\}$ is a submodule of $M$ that contains $N$.
The bijection $\vartheta$ sends $S \mapsto q^{-1}(S)$.