Artinian module

A module satisfying the descending chain condition on submodules.

Artinian module

An $R$ -module $M$ is Artinian if it satisfies the descending chain condition (DCC) on submodules : for every chain

N_1 \supseteq N_2 \supseteq N_3 \supseteq \cdots

there exists $k$ such that $N_k=N_{k+1}=\cdots$ . For many classes of modules, Artinianity is equivalent to having finite length; compare length .

Artinian modules are “finite from below” in their submodule lattice and are the setting for induction on minimal submodules.

Examples:

Any finite abelian group is Artinian as a $\mathbb Z$ -module.
Any finite-dimensional vector space over a field is Artinian (every descending chain of subspaces stabilizes).
(Nonexample) $\mathbb Z$ is not Artinian: the chain $\mathbb Z \supset 2\mathbb Z \supset 4\mathbb Z \supset \cdots$ never stabilizes.