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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?

Maths Wizzard
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1 Answers1

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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 $S \mapsto q^{-1}(S)$.

Kenny Wong
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