Corollary 2

Suppose that $A$ and $B$ are Noetherian and that $B$ is a flat $A$-module. Let $p$ be a prime ideal of $A$, $Q \subset E$ a $p$-primary submodule and $\mathfrak{P}$ a prime ideal of $B$. For $Q \otimes_A B$ to be a $q$-primary submodule of $E \otimes_A B$, it is necessary and sufficient that $pB$ be a 'p-primary ideal' of $B$.

Let us apply Theorem 2 to the $A$-module $E/Q$ and the $B$-module $B$; then $\operatorname{Ass}_A(E/Q) = \{p\}$ and $(E/Q) \otimes_A B$ is isomorphic to $(E \otimes_A B)/(Q \otimes_A B)$ and hence $\operatorname{Ass}_B((E \otimes_A B)/(Q \otimes_A B)) = \operatorname{Ass}_B(B/pB)$. To say that $Q \otimes_A B$ is $\mathfrak{P}$-primary in $E \otimes_A B$ therefore means that $\operatorname{Ass}_B(B/pB)$ is reduced to $\mathfrak{P}$, whence the corollary.