Artinian
From AoPSWiki
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.
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.
