Noetherian ring

Definition

Let $R$ be a commutative ring. The ring $R$ is Noetherian if the set of its ideals, ordered by inclusion, satisfies the ascending chain condition. That is, for every chain $I_1 \subseteq I_2 \subseteq \dots$ of ideals of $R$, there is $N$ with $I_n = I_N$ for all $n \ge N$.

Remarks

The ring $R$ is Noetherian if and only if every ideal of $R$ is finitely generated. Every field and every principal ideal domain is Noetherian. By Hilbert's basis theorem, $R[x_1, \dots, x_n]$ is Noetherian when $R$ is. The polynomial ring in infinitely many variables $x_1, x_2, \dots$ over a field is not Noetherian: $(x_1) \subset (x_1, x_2) \subset \dots$ never stops. For a noncommutative ring, one defines left and right Noetherian rings with left and right ideals.

Explanations

L1 human

In a Noetherian ring, each ideal is generated by a finite list of elements. This makes many arguments work by induction, because a growing chain of ideals must stop. Rings met in practice, like $\mathbb{Z}[x, y]$, are Noetherian.