Noetherian ring

Let $R$ be a commutative ring.

Definition

$R$ is Noetherian if it satisfies the ascending chain condition (ACC) on ideals : every chain

stabilizes (i.e., $I_n=I_{n+1}=\cdots$ for $n\gg 0$).

Equivalently (and very useful in practice): every ideal of $R$ is finitely generated.

Standard permanence properties

• If $R$ is Noetherian and $I\subseteq R$ is an ideal, then $R/I$ is Noetherian.
• If $R$ is Noetherian, then $R[x]$ is Noetherian (Hilbert basis theorem ); hence so is $R[x_1,\dots,x_n]$.
• If $R$ is Noetherian and $S$ is a multiplicative set , then the localization $S^{-1}R$ is Noetherian (localization is Noetherian ).

Examples

1. Basic examples.
   Any field is Noetherian, and $\mathbb{Z}$ is Noetherian (in fact a PID ).

2. Polynomial rings in finitely many variables.
   If $k$ is a field, then $k[x_1,\dots,x_n]$ is Noetherian by the Hilbert basis theorem.

3. A non-example (infinitely many variables).
   The ring $k[x_1,x_2,x_3,\dots]$ is not Noetherian: the chain

   $$(x_1)\subset (x_1,x_2)\subset (x_1,x_2,x_3)\subset \cdots$$

   never stabilizes.