Corollary 2

Let A, B be two commutative rings, $p : A \to B$ a homomorphism such that B is a flat A-module and E an A-module. Then, for every finitely generated submodule F of E, the annihilator of $F_{(B)}$ is the ideal $aB$ of B, where a is the annihilator of F in A.

It suffices to apply Proposition 12 with $E_1 = A$, $E_2 = E_3 = E$, $F_2 = F$, $F_3 = \{0\}$.