Artinian module - Knowlpedia

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

N_1 \supseteq N_2 \supseteq N_3 \supseteq \cdots

there exists $k$ such that $N_k=N_{k+1}=\cdots$ .

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

Relation to finite length

Every module of finite length is Artinian. The converse does not hold for arbitrary rings and modules.

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.