Noetherian ring

Last updated January 27, 2022

Definition

Rings

Definition. A Noetherian ring is associative, commutative ring with identity $A$ such that one of the following equivalent conditions hold:

1. Every ideal $I\subset A$ is finitely generated, i.e. $\exists f_1,\dots,f_n\in I$ such that for all $a\in I$, $a=\sum a_i f_i$ where $a_i\in A$.
2. (Ascending chain condition) For all infinite chains $$ I_1\subset I_2\subset\cdots\subset I_k\subset\cdots $$ there exists $N$ such that $I_N=I_{N+1}=\cdots$.

Proposition. The conditions above are indeed equivalent.

Modules

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

Definition. $M$ is Noetherian if one of the following equivalent conditions hold:

1. Every ascending chain of submodules stabilizes.
2. Any submodule of $M$ is finitely generated.

Properties

1. If $A$ is Noetherian and $I\subset A$ is some ideal, then $A/I$ is Noetherian.
2. Hilber basis theorem.. If $A$ is a Noetherian ring, then so is $A $.
3. If $A$ is a Noetherian ring then $A^n$ is a Noetherian module.
4. Lemma
   1. Submodule of a Noetherian module is Noetherian
   2. Let $\phi:M\to N$ be a surjective module homomorphism. Then if $M$ is Noetherian then $N$ is Noetherian.
   3. Given a sequence $$0\to N\to M\to M/N\to 0$$ if $N$ and $M/N$ are Noetherian then so is $M$.
   4. A finitely generated module $M$ of a Noetherian ring $A$ is also Noetherian.
5. If $A$ is Noetherian then $A[ ]$ is Noetherian (formal power series).
6. Nilradical is nilpotent (since it is f.g.)

Examples

• Any PID
• Any field $K$
• For a Noetherian ring $A$, $A $ is a Noetherian ring

Dimension

• dimension 0: Artinian ring
• dimension 1: includes dedekind domain