Artinian ring

Let $R$ be a commutative ring .

Definition

$R$ is Artinian if it satisfies the descending chain condition (DCC) on ideals: for every chain

there exists $N$ such that $I_n=I_N$ for all $n\ge N$.

Equivalently, every nonempty set of ideals of $R$ has a minimal element under inclusion.

Key consequences in the commutative case

• Every commutative Artinian ring is Noetherian . In particular, many finiteness tools become available automatically.
• A commutative Artinian ring has Krull dimension $0$: every prime ideal is maximal. In terms of spectra, $\operatorname{Spec}(R)$ is a finite discrete space and coincides with the maximal spectrum .
• The Jacobson radical (the intersection of maximal ideals ) is nilpotent, and $R$ decomposes as a finite product of Artinian local rings; this is closely related to the Chinese remainder theorem .

Examples

1. Fields.
   Any field is Artinian: its only ideals are $(0)$ and $(1)$.

2. Finite quotients of $\mathbb{Z}$.
   $\mathbb{Z}/n\mathbb{Z}$ is Artinian (it is finite, so it has only finitely many ideals). For instance, $\mathbb{Z}/12\mathbb{Z}$ is Artinian.

3. Truncated polynomial rings.
   If $k$ is a field, then $k[x]/(x^n)$ is Artinian: the ideals are $(0)\subset (x^{n-1})\subset \cdots \subset (x)\subset (1)$, so every descending chain stabilizes.

(As a contrast, $k[x]$ is Noetherian but not Artinian: the descending chain $(x)\supset (x^2)\supset (x^3)\supset \cdots$ never stabilizes.)