\begin{definition}[Definition:Artinian Module]
Let $A$ be a commutative ring with unity.
Let $M$ be an $A$-module
Then $M$ is a '''Artinian module''' if either of the following conditions hold:
:$(1): \quad$ $M$ satisfies the descending chain condition on submodules
:$(2): \quad$ $M$ satisfies the minimal condition on submodules.
\end{definition}