Let $A \subset B$ be an inclusion of commutative rings inducing $f: \text{Spec}(B) \rightarrow \text{Spec}(A)$. Must it be the case that $\text{Ass}(A) \subset f(\text{Ass}(B))$?
If this isn't true in general, is it true when $A,B$ are Noetherian rings? How generally can we do it?
For example, if $B$ is integral over $A$, then $\text{Spec}(B) \rightarrow \text{Spec}(A)$ is surjective. If $\mathfrak{p}$ is an associated prime of $A$, then write $\mathfrak{p} = \text{ann}(s)$. This is the image of some prime $\mathfrak{q} \subset B$, and an associated prime obtained from extending $\text{ann}(s)$ in $B_{\mathfrak{q}}$ should produce an associated prime which maps to $\mathfrak{q}$.