# 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 $A$ be an Artinian ring. Then every prime ideal of $A$ is maximal. Thus $\dim A = 0$ (the Krull dimension of $A$ is $0$).

Theorem. Let $A$ be a ring. Then $A$ is Artinian if and only if $A$ is Noetherian and every element of $A$ is either invertible or nilpotent.

However, Artinian modules are not necessarily Noetherian. Consider, for example, the Prüfer Group for some prime $p$ as a $\mathbb{Z}$-module (i.e., the additive group of rationals of the form $a/p^k$, modulo $\mathbb{Z}$). Each of its submodules is of the form $(1/p^n)$, for some integer $n \ge 0$. 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 $\[(1/p^0) \supset (1/p^1) \supset (1/p^2) \supset \dotsb .\]$ This module is therefore Artinian, but not Noetherian.