It is well known as Cohen's theorem that a commutative ring is Noetherian if all its
prime ideals are finitely generated. Is this statement true or false when prime ideals are replaced by maximal ideals? 



A remark on Cohen's theorem
See the following paper (and search for its citations for related work)


Gilmer, R; Heinzer W.  

A non-Noetherian two-dimensional Hilbert domain with principal maximal ideals,  

Michigan J. Math. 23 (1976), 353-362  

[Link](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-23/issue-4/A-non-Noetherian-two-dimensional-Hilbert-domain-with-principal-maximal/10.1307/mmj/1029001770.full)


