A ring is called left (respectively right) Artinian if it does not contain an infinite descending chain of left (resp. right) ideals. In this case the ring in question is said to satisfy the descending chain condition on left (resp. right) ideals. A ring is called Artinian if it is both left and right Artinian.

Artinian module | descending chain condition | left ideal | Noetherian ring | right ideal

Emil Artin