# Artinian module

A module that satisfies the descending chain condition for submodules. The class of Artinian modules is closed with respect to passing to submodules, quotient modules, finite direct sums and extensions. Extension in this context means that if the modules $B$ and $A/B$ are Artinian, then so is $A$. Each Artinian module can be decomposed into a direct sum of submodules which are no longer decomposable into a direct sum. A module has a composition series if and only if it is both Artinian and Noetherian. See also Artinian ring.