Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then $M$ is Noetherian is equivalent to $M'$ and $M''$ are Noetherian.
For the ''$\Leftarrow$'' case: I guess if we let $(L_n)_{n\geq 1}$ be an ascending chain of submodules of $M$, then $(\alpha ^{-1}(L_n))_{n\geq 1}$ is a chain in $M'$, and $(\beta(L_n))_{n\geq 1}$ is a chain in $M''$. For large $n$ both these chains are stationary. Then why we know the chain $(L_n)$ is stationary?
Thanks!