Artinian module

An $R$-module is Artinian when every descending chain of submodules stabilizes. Equivalently, every nonempty family of submodules has a minimal member.

For an injective module endomorphism $f:M\to M$ of an Artinian module, the chain $M\supseteq fM\supseteq f^{2}M\supseteq \cdots$ stabilizes. If $f^{n}M=f^{n+1}M$, then every $x$ satisfies $f^{n}x=f^{n+1}y$. Injectivity of $f^n$ gives $x=fy$, proving surjectivity. No finite-length or Noetherian hypothesis is needed.

For a short exact sequence $0\to N\to M\to Q\to 0$, 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.