If the start and end of an exact sequence are both Artinian modules, then is the middle one also an Artinian module?

Question

$A\to B \to C$ is an exact sequence. If A and C are Artinian modules, is B also an Artinian module?

I think it is wrong. But I failed to find any counterexample.

Answer 1

Would you agree that

$0\to \mathrm{Im}(f)\overset{\subseteq}\longrightarrow B\overset{g}\longrightarrow\mathrm{Im}(g)\to 0$

is a short exact sequence, where $g$ is abused to denote both functions $B\to C$ and $B\to \mathrm{Im}(g)$?

Notice that $\mathrm{Im}(f)$ is a homomorphic image of an Artinian module, and $\mathrm{Im}(g)$ is a submodule of an Artinian module.

Considering that, how do you feel about $B$ being Artinian?

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Maybe it's the other direction you should be suspicious of (meaning a claim that if $B$ is Artinian, then both $A,C$ are too.) If $S$ is a simple module, then there are many such sequences like this

$$0\to S\to \oplus_{n\in\mathbb N}S$$

$$\oplus_{n\in\mathbb N}S\to S\to 0$$

that are exact, and $S$ is Artinian, of course.