Noetherian ring

The following are equivalent:

1. $R$ is Noetherian (ACC on ideals).
2. Every ideal of $R$ is finitely generated.
3. Every nonempty set of ideals of $R$ has a maximal element under inclusion.

The equivalence of (1) and (2) is the most used in practice.

• If $R$ is Noetherian and $I\subseteq R$ is an ideal, then $R/I$ is Noetherian.
• If $R$ is Noetherian and $S$ is a multiplicative set, then the localization $S^{-1}R$ is Noetherian; see localization preserves Noetherianity.
• If $R$ is Noetherian, then $R[x_1,\dots,x_n]$ is Noetherian (Hilbert basis theorem); a common formulation appears as Hilbert basis corollary.

Noetherian hypotheses are the natural setting for finiteness results such as primary decomposition via the Lasker–Noether theorem.

1. Basic arithmetic and algebra. $\mathbb{Z}$ is Noetherian (every ideal is principal). If $k$ is a field, then $k[x_1,\dots,x_n]$ is Noetherian.

1. Quotients and finitely generated algebras. If $R$ is Noetherian, then so is $R/I$ for any ideal $I$, and so is any finitely generated $R$-algebra of the form $R[x_1,\dots,x_n]/J$.

1. Localizations. $\mathbb{Z}_{(p)}$ (localization of $\mathbb{Z}$ at the prime $(p)$, i.e. inverting all integers not divisible by $p$) is Noetherian by localization and the permanence above.

• The polynomial ring in countably many variables $k[x_1,x_2,x_3,\dots]$ is not Noetherian: the chain of ideals
  $$(x_1) \subset (x_1,x_2) \subset (x_1,x_2,x_3) \subset \cdots$$
  never stabilizes.