= Noetherian modules in a short exact sequence
{c}
{title2=$M\text{ Noetherian}\iff N,\ M/N\text{ Noetherian}$}
For a <short exact sequence> $0\to N\to M\to M/N\to0$, $M$ is a <Noetherian module> exactly when both the other terms are <Noetherian modules>. For the extension direction, an ascending chain $M_i$ has eventually constant intersections $M_i\cap N$ and images $(M_i+N)/N$. Equality of both forces equality of consecutive $M_i$: subtract an element with the same image and then use the common intersection.
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