Assume that $ B \hookrightarrow A $ is an injective ring homomorphism.

Question:If $\mathfrak{p}$ is an associated prime of $B$, is there an associated prime $ \mathfrak{q}$ of $A$ such that $ \mathfrak{q} \cap B = \mathfrak{p}$?

Some comments:

- This is true if $ \mathfrak{p}$ is a minimal prime of $B$. Just take an associated prime of $ (B \backslash \mathfrak{p})^{-1} A$. So the only problem is with embedded primes
- The question is an algebraic translation of the following more geometric question: If $X \to Y$ is a quasicompact morphism of noetherian schemes, is every associated point of the schematic image hit by some associated point of $X$?