Artinian ring

Last updated January 27, 2022

Definition

Rings

Let $A$ be a commutative ring.

Definition. $A$ is called Artinian if every ascending chain of ideals stabilizes, i.e. for every chain is of the form $$ J_1\supseteq J_2\supseteq\dots\supseteq J_i\supseteq\cdots $$ where $J_i=J_{i+1}=\dots$.

Modules

Let $M$ be an $A$-module.

Definition. $M$ is Artinian if every ascending chain of submodules stabilizes.

Properties

1. Every Artinian ring is Noetherian. (The converse may fail.)
2. Artinian rings are Noetherian rings of dimension 0.
   1. This is equivalent to saying every prime ideal is maximal.
3. Lemma
   1. Submodule of a Artinian module is Artinian
   2. Let $\phi:M\to N$ be a surjective module homomorphism. Then if $M$ is Artinian then $N$ is Artinian.
   3. Given a sequence $$0\to N\to M\to M/N\to 0$$ if $N$ and $M/N$ are Artinian then so is $M$.
4. The Jacobson radical is nilpotent.
5. There are finitely many maximal ideals

Examples

1. Fields
2. $k /(f_1^{n_1}\cdots f_k^{n_k})$
   1. $k /(X^n)$
3. $\mathbb{Z}/n\mathbb{Z}$