Cohen's theorem on Noetherian rings

Remarks

One direction is clear, since in a Noetherian ring every ideal is finitely generated. For the other, suppose that some ideal is not finitely generated. By Zorn's lemma there is an ideal $I$ that is maximal among the ideals that are not finitely generated. Let $a, b \notin I$ with $ab \in I$. Then $I + (a)$ and the ideal quotient $(I : a) = \set{r \in R : ra \in I}$ are strictly larger than $I$, hence finitely generated. One checks that $I$ would then be finitely generated, which is impossible; so $I$ is prime. This prime ideal is not finitely generated, against the hypothesis. For example, to show that $\Z[x]$ is Noetherian it would be enough to check its prime ideals, although Hilbert's basis theorem gives this more directly.

Going deeper

For a student

The proof is a model of a common method: an ideal maximal for not having a property is often prime. The same method shows that a commutative ring in which every prime ideal is principal is a principal ideal ring. The theorem is a test for the Noetherian property, for example for rings of power series and rings of integers.