A formal consequence of Krull's principal ideal theorem is the following:
If $A$ is a Noetherian ring, and $I$ is an ideal generated by $r$ elements, then any prime ideal which is minimal among those that contain $I$ has height at most $r$.
This statement implies that for Noetherian rings, principal prime ideals have height at most $1$.
My question is if this is true for any ring, i.e., is a principal prime ideal of a ring always of height at most $1$?
The above question is clearly true if the following statement is true: If every maximal ideal of a ring is finitely generated, then the ring is Noetherian. (Note that this statement is true if we replace maximal ideals by prime ideals)
However, I am not sure if this latter assertion is true, although I do not have a counterexample for it.