# Artinian

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.