I proved part (i) of the following:
Proposition 6.3. Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then
i) $M$ is Noetherian $\iff$ $M'$ and $M'"$ are Noetherian;
ii) $M$ is Artinian $\iff$ $M'$ and $M''$ are Artinian
Proof. We shall prove i); the proof of ii) is similar.
$\implies$: An ascending chain of submodules of $M'$ (or $M''$) gives rise to a chain in $M$, hence is stationary.
Can you tell me if my proof of $(i)\Longleftarrow$ is correct? Here goes:
Let $M^\prime$ and $M^{\prime \prime}$ be Noetherian. Let $L_n$ be an ascending chain of submodules in $M$. Then $\alpha^{-1}(L_n)$ is an ascending chain of submodules in $M^\prime$. Hence $\alpha^{-1}(L_n)$ is stationary, that is, $\alpha^{-1}(L_n) = \alpha^{-1}(L_{n+1})$ for $n$ large enough. Hence $L_n = L_{n+1}$ for $n$ large enough since $\alpha$ is injective hence $L_n$ is stationary.
The proof given in Atiyah-Macdonald is the following but I don't understand why they need both maps, $\alpha$ and $\beta$:
$\Longleftarrow$: Let $(L_n)_{n \ge 1}$ be an ascending chain of submodules of $M$; then $(\alpha^{-1}(L_n))$ is a chain in $M'$, and $(\beta(L_n))$ is a chain in $M''$. For large enough $n$ both these chains are stationary, and it follows that the chain $(L_n)$ is stationary. $\Box$