A module $M$ is artinian if it satisfies the following equivalent conditions:

• the descending chain condition holds for submodules of $M$ ;
• every nonempty family of submodules of $M$ has a minimal element.

A ring $R$ is left artinian if it is artinian as a left module over itself (i.e. if $_RR$ is an artinian module), and right artinian if it is artinian as a right module over itself (i.e. if $R_R$ is an artinian module), and simply artinian if both conditions hold.