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?

See the following paper (and search for its citations for related work) Gilmer, R; Heinzer W. 


Here is another nice ring which shows that the answer is: "false". It is due to B. Osofsky and can serve as (counter)example in other situations. Start with the $p$adics $\mathbb{Z}_p$ and $\mathbb{Q}_p$. (Any other complete DVR with its fraction field would do.) Our ring, call it $R$, is the additive group $\mathbb{Z}_p \oplus \mathbb{Q}_p/\mathbb{Z}_p$ with multiplication $(a,b) \cdot (c,d) = (ac, ad + cb)$. (This is called the "trivial extension" of the ring $\mathbb{Z}_p$ by its module $\mathbb{Q}_p/\mathbb{Z}_p$.) The ideals in $R$ form the chain $R \supsetneq pR \supsetneq p^2R \supsetneq \; ... \; (0, \mathbb{Q}_p/\mathbb{Z}_p) \; ... \; \supsetneq (0, p^{2}\mathbb{Z}_p/\mathbb{Z}_p) \supsetneq (0, p^{1}\mathbb{Z}_p/\mathbb{Z}_p) \supsetneq 0$ Among other interesting properties that $R$ has (like being a cogenerator ring, in particular selfinjective), it is local with principal maximal ideal $pR$; but it has exactly one nonmaximal prime ideal  the lonely one there in the middle of the chain, having no neighbours , and this sad thing is not finitely generated. 

