= Artinian modules in a short exact sequence
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For a <short exact sequence> $0\to N\to M\to Q\to0$, the middle <module> is <Artinian> exactly when both outer <modules> are <Artinian>. Submodules and quotients inherit the <descending chain condition>. Conversely, for a descending chain $M_i$, both $M_i\cap N$ and its image in $Q$ eventually stabilize. If $x\in M_i$, choose $y\in M_{i+1}$ with the same image in $Q$; then $x-y\in M_i\cap N=M_{i+1}\cap N$, giving $M_i=M_{i+1}$. Induction shows that finite direct sums of <Artinian modules> are <Artinian>.
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