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



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> >  **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}$?
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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 $X$?



Lying over theorem for associated primes
Let me explain why the statement is true. Let $i:B\to A$ be an injection of noetherian rings and $\mathfrak{p}$ be an associated prime of $B$ : it is the annihilator of a nonzero element $f\in B$.


Let us first localize at $\mathfrak{p}$. More precisely, replacing $B\to A$ by $B\_{\mathfrak{p}}\to A\otimes\_B B\_{\mathfrak{p}}$ , it is possible to assume that $B$ is local with maximal ideal $\mathfrak{p}$.


Let $I$ be the ideal of $A$ generated by $i(f)$, viewed as an $A$-module. Since $I\neq 0$ by injectivity of $i$, it has an associated prime $\mathfrak{q}\subset A$. This means that there is $a\in A$ such that $\mathfrak{q}$ is the annihilator of ai(f). Note that $\mathfrak{q}$ is also associated to $A$, and let us check that it has the required properties. As $\mathfrak{p}$ annihilates $f$, $\mathfrak{q}\cap B$ contains $\mathfrak{p}$. Since $\mathfrak{p}$ is maximal, $\mathfrak{q}\cap B=\mathfrak{p}$.


