Artinian module

Remarks

Equivalently, every nonempty set of submodules of $M$ has a minimal element for inclusion. Example: a finite-dimensional vector space $V$ over a field is Artinian, because a strictly decreasing chain of subspaces has at most $\dim V + 1$ terms. Non-example: $\Z$ as a $\Z$-module is not Artinian, because $2\Z \supsetneq 4\Z \supsetneq 8\Z \supsetneq \cdots$ never stops. Example: for a prime $p$, the Prüfer group $\Z[1/p]/\Z$ is an Artinian $\Z$-module, since its proper subgroups form the chain $0 \subset \frac{1}{p}\Z/\Z \subset \frac{1}{p^2}\Z/\Z \subset \cdots$. That chain also shows that it is not a Noetherian module. Submodules, quotients and extensions of Artinian modules are Artinian.