Noetherian ring

Noetherian rings are important structures that arise in algebraic-geometric settings.

Definition

A ring is left (right) Noetherian if every left (right) ideal is finitely generated. It is called Noetherian if it is both left and right Noetherian. Note this is true of any left or right Noetherian ring in the commutative case.

Equivalent definition

Let $R$ be a ring. $R$ is (left, right) Noetherian if and only if every chain of (left, right) ideals stabilizes. That is, if

$$I_1\subset I_2\subset \cdots \subset R$$

is a chain of ideals then $I_n=I_{n+1}=\cdots$ for some $n$.

Proof

This proof is only for the commutative case, but the same idea applies.

Consider a chain of ideals

$$I_1\subset I_2\subset \dots$$

Then $I={\bigcup }_{j=1}^{\infty }{I}_j$ is an ideal (it’s kind of like the limit of the chain of ideals) where $I=(r_1,\dots ,r_m)$ for some $r_1,\dots r_m\in R$. So there are only $m$ generators to work with so, as soon as $r_1,\dots r_m\in I_n$ then $I_n=I_{n+1}=\dots$ since they all have the same generators.

Going the other way, assume that $R$ satisfies the ascending chain condition. Let $I=(r_1,\dots )$ be an ideal. Then we have the chain

$$(r_1)\subset (r_1,r_2)\subset \cdots$$

which stabilizes by assumption, thus after some point adding generators doesn’t do anything so $I$ is finitely generated.

Interesting related theorem/ ideas

• Hilbert Basis Theorem