Being Noetherian Is Not Local…Or Is It?

A commutative ring $R$ can be non-Noetherian and have all of its localisations at prime ideals Noetherian, such as the infamous $\prod_{i=1}^\infty \mathbb{Z}/2$. So being Noetherian is not a local property. However, there is an interesting variant of 'local' that does work, which I learnt from Yves Lequain's paper [1]. It goes like this:

Theorem. Let $R$ be a ring and fix a left maximal ideal $M$ of $R$. Then $R$ is left Noetherian if and only if every left ideal contained in $M$ is finitely generated.

The nice thing about this statement is that it avoids localisation so it's easy to state for noncommutative rings.

Proof. The 'only if' direction follows directly from the definition. Now suppose that every left ideal contained in $M$ is finitely generated. Let $I$ be an arbitrary left ideal of $R$. Our goal is to show that $I$ is finitely generated.

To do this, observe that because $M$ is maximal, the $R$-module homomorphism $I\oplus M\to R$ given by $(i,m)\mapsto i + m$ is actually surjective, and its kernel is isomorphic to $I\cap M$. Hence, there is a short exact sequence of left $R$-modules
$$0\to I\cap M\to I\oplus M\to R\to 0.$$
However, $R$ is projective, so this sequence is split and we get an isomorphism of left $R$-modules
$$I\oplus M\cong R\oplus (I\cap M).$$
Since $R$ is finitely generated and $I\cap M$ is finitely generated by hypothesis, $I\oplus M$ is also finitely generated and hence $I$ is as well.

It's well-known that if every prime ideal in a commutative ring is finitely generated, then every ideal is since an ideal maximal with respect to being not finitely generated is actually prime. However, Lequain in his paper provides an example of a commutative ring $R$ and a maximal ideal $M$ such that every prime in $M$ is finitely generated, but that $R$ is still not Noetherian. Crazy!

I invite the reader to look at Lequain's short 2-page paper [1] where he also proves that a commutative ring $R$ is a Dedekind domain if and only if every nonzero ideal contained in $M$ is invertible (here, $M$ is still some fixed maximal ideal).

[1] Lequain, Yves. A local characterization of Noetherian and Dedekind rings. Proc. Amer. Math. Soc. 94 (1985), no. 3, 369-370.