Corollary 3

Let $A$ be a commutative ring, $a$ an ideal of $A$ and $E$ and $F$ two $A$-modules such that $a$ is contained in the annihilator of $F$. Then the $(A/a)$-modules $E \otimes_A F$ and $(E/aE) \otimes_{A/a} F$ are canonically isomorphic.

$F$ and $E \otimes_A F$ are annihilated by $a$ and hence have canonical $(A/a)$-module structures ($§ 1$, no. 12) and if we write $E' = aE$, then $\operatorname{Im}(E' \otimes_A F) = 0$; then there is a canonical isomorphism (20) of $E \otimes_A F$ onto $(E/aE) \otimes_A F$ and the latter is itself identical with $(E/aE) \otimes_{A/a} F$ (no. 3, Corollary to Proposition 2).