Artinian ring

Let $R$ be a commutative ring.

Definition

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

stabilizes.

Useful facts (commutative case)

• Every Artinian ring is Noetherian .
• $\mathrm{Spec}(R)$ is finite and every prime ideal is maximal ; equivalently the Krull dimension is $0$.
• Artinian rings often split as finite products via Chinese remainder ideas, especially when maximal ideals are comaximal.

Examples

1. Fields.
   A field has only two ideals $(0)$ and $(1)$, so it is Artinian.

2. $\mathbb{Z}/n\mathbb{Z}$.
   The ring $\mathbb{Z}/n\mathbb{Z}$ is Artinian (in fact finite), hence also Noetherian.

3. A local Artinian example.
   $k[x]/(x^n)$ is Artinian and local with maximal ideal generated by the class of $x$; that class is nilpotent .