Is a prime ideal lying between associated primes also an associated prime?

Question

Let $A$ be a ring and consider three prime ideals in $A$ with inclusions $P\subset Q\subset P'$. Suppose that $P,P'\in \operatorname{Ass}(A)$. Must then also $Q\in \operatorname{Ass}(A)$?

Answer 1

It seems not. E.g., let $k$ be a field, $A=k[x,y,z]/(z^2,xz,yz)$ (plane with embeddeded point). The associated primes are $(z)$ and $(x,y,z)$, but the intermediate ideal $(x,z)$ is not.