Proposition 6

Let E be a right A-module and F a left A-module such that E is F-flat. For every submodule F' of F, we denote by \phi(F') the image of E \otimes F' under the canonical mapping of E \otimes F' to E \otimes F (which is injective by Definition 1 of no. 2). Then, if F', F'' are two submodules of F, we have

As E is F-flat, $\phi(F' + F'')$ is identified with $E \otimes (F' + F'')$, and the submodules $\phi(F')$, $\phi(F'')$ and $\phi(F' \cap F'')$ are identified with the canonical images of $E \otimes F'$, $E \otimes F''$ and $E \otimes (F' \cap F'')$ in $E \otimes (F' + F'')$ respectively. Proposition 6 then follows from Lemma 7.