TITLE: Rings with all non-prime ideals finitely generated
QUESTION [9 upvotes]: Motivated by this question, 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.

REPLY [8 votes]: 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.