We say that a ring or module is Artinian if the descending chain condition holds for its ideals/submodules. The notion is similar to that of Noetherian rings and modules.
One might expect Artinian rings to be just as broad and diverse a category as Noetherian rings. However, this is not the case. In fact, Artinian rings are Noetherian but the converse does not hold.
Theorem. Let be an Artinian ring. Then every prime ideal of is maximal. Thus (the Krull dimension of is ).
Theorem. Let be a ring. Then is Artinian if and only if is Noetherian and every element of is either invertible or nilpotent.
However, Artinian modules are not necessarily Noetherian. Consider, for example, the Prüfer Group for some prime as a -module (i.e., the additive group of rationals of the form , modulo ). Each of its submodules is of the form , for some integer . Thus a descending chain of submodules corresponds uniquely to an increasing sequence of nonnegative integers, and vice-versa. Thus every ascending chain must stabilize, but we have the descending chain This module is therefore Artinian, but not Noetherian.