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Firstly: Am I interpreting the correspondence theorem for modules correctly? If I have $B$ a submodule of $A$, then the submodules of $A/B$ correspond to the submodules of $A$ that contain $B$?

Now if that is the case, it seems then that if $C$ is a nontrivial submodule of $A/B$ then $C$ contains $B$ and $A/B/C$ is well defined. But else if $C$ does not contain $B$, then $C$ is not a submodule of $A/B$ is that correct?

Secondly: Would it be correct to say that $A/B/C=A/C/B$. In my mind, it seems that for $A/B/C$ to be well defined, $C$ must be a submodule of $A/B$ and hence $C$ contains $B$, and hence $A/C/B=A/C$, is that correct?

But then to my confusion, say $A$ is a $3$-dim $R$-module generated by $\{a,b,c\}$ and $B=\langle \{b\}\rangle, C=\langle \{c\}\rangle$. Then $A/B$ has generators $\{a+B,c+B\}$ and the submodule of this generated by $\{c+B\}$ corresponds to a submodule of $A$ that does not contain $B$.

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Now if that is the case, it seems then that if $C$ is a nontrivial submodule of $A/B$ then $C$ contains $B$ and $A/B/C$ is well defined.

No, it does not make sense to say a submodule of $A/B$ contains $B$. The theorem does not say "the submodules of $A/B$ are submodules of $A$": the bold text ought to read correspond to.

That means that there is a submodule $C'$ lying between $A$ and $B$ such that $C'/B=C$.

$C$ is a subset of $A/B$, not $A$. You should definitely not consider comparing such entities with containment.

"$A/B/C$" is quite unclear. It would be better to write $(A/B)/C$ which is defined, yes. Alternatively you could express $C$ as $C'/B$ as above, and write $(A/B)/(C'/B)$ which, by the third isomorphism theorem, is isomorphic to $A/C'$.

$A/B/C=A/C/B$

The meaning is totally unclear because the reader can't tell what is a quotient of what. I already discussed how you can express $A/B/C$ correctly above. Perhaps what you want to get at on the right side is $A/C'\cong (A/B)/(C'/B)$ as also explained above. Otherwise no interpretation of it makes sense: $C$ isn't a subset of $A$, it isn't a superset of $B$, so $A/C$ and $C/B$ are both nonsensical.

Draw pictures

Maybe if we pretend we're looking at the lattice of submodules of $A/B$ your picture will get clearer. Of course the biggest submodule is $A/B$ itself, and the smallest submodule is $B/B$. We are just saying that whenever you pick out a submodule $C$

$$ B/B\subseteq C\subseteq A/B $$

then up in the submodule lattice of $A$, there is a $C'$ such that

$$ \{0\}\subseteq B\subseteq C'\subseteq A$ $$

such that you can replace $C$ with $C'/B$:

$$ B/B\subseteq C'/B=C\subseteq A/B $$

rschwieb
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