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

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

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