Is there something missing from the highlighted statement?
From my understanding, if a ring is Noetherian, then it is finitely generated.
2026-03-26 11:17:21.1774523841
Examples of rings that are Noetherian but not finitely generated?
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The misunderstanding is about the usage of “finitely generated”.
The part you emphasized is about Noetherian rings which are algebras over a field, but are not finitely generated as algebras.
An algebra $R$ over a field $F$ is finitely generated as an algebra if (and only if) there exists a surjective $F$-algebra homomorphism $F[X_1,X_2,\dots,X_n]\to R$, for some $n$. Since the domain is Noetherian, the codomain is Noetherian as well.
On the other hand, an $F$-algebra can be Noetherian without being finitely generated as an algebra: take the algebraic closure of a finite field, for instance, or $\mathbb{C}$ as an algebra over $\mathbb{Q}$.
An example related to the comment in the quoted text is the localization $\mathbb{Z}_{(p)}$ at a prime $p$. This is a Noetherian ring, but it is not finitely generated, because there are infinitely many primes.