Motivated by [this question](https://mathoverflow.net/questions/302468/local-ring-all-whose-non-maximal-ideals-are-finitely-generated), I would like to ask: 



> 
> If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
> 
> 
> 


Note that for zero-dimensional rings, the answer is yes by the linked question. So we only need to think about rings of positive dimension. 


NOTE: All our rings are commutative with unity.



Rings with all non-prime ideals finitely generated
No.


If $\mathbb Z\_{p^{\infty}}$ is a Prufer group for prime $p$, then its endomorphism ring is isomorphic to the ring $\mathbb Z\_p$ of $p$-adic integers. Hence $\mathbb Z\_{p^{\infty}}$ is a $\mathbb Z\_p$-module, and we can form the idealization $R:=\mathbb Z\_p\oplus \mathbb Z\_{p^{\infty}}$. The ideals of this ring are known: they are of the form $0\oplus H$ for a subgroup $H\leq \mathbb Z\_{p^{\infty}}$ or of the form $(p^k)\oplus \mathbb Z\_{p^{\infty}}$ for some nonnegative integer $k$. 
(See Example 1 of Paul A. Froeschl, Chained rings. Pacific J. Math.
Volume 65, Number 1 (1976), 47-53.)
In particular, $R$ is a local ring with exactly one non-finitely generated ideal $I:=0\oplus \mathbb Z\_{p^{\infty}}$. This ideal is prime because $R/I$ is isomorphic to $\mathbb Z\_p$, a domain.


