artinian

A module $M$ is artinianif 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 artinianif it is artinian as a left module over itself(i.e. if ${}_{R}R$ is an artinian module),and right artinianif 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.