Commutative rings with every non-maximal ideal finitely generated

TL;DR

This paper investigates the structure of commutative rings where all non-maximal ideals are finitely generated, proving that such rings are necessarily Noetherian with all ideals finitely generated.

Contribution

It establishes that if all non-maximal ideals in a commutative ring are finitely generated, then the ring is Noetherian, extending Cohen's theorem.

Findings

All non-maximal ideals finitely generated implies ring is Noetherian.

Maximal ideals are finitely generated under this condition.

The ring's ideal structure is fully finitely generated, confirming Noetherian property.

Abstract

In commutative ring theory, there is a theorem of Cohen which states that if in a commutative ring all prime ideals are finitely generated then every ideal is finitely generated. However, it is known that having only maximal ideals finitely generated doesn't imply all ideals be finitely generated. In his article we ask the question that what happens if we assume all non-maximal ideals are finitely generated, and we answer the question by showing that indeed then all maximal ideals are also finitely generated i.e. the ring becomes Noetherian.