Proposition 10

Let $B$ be a ring and $A$ a subring of $B$ such that $B$ is a faithfully flat right $A$-*module. Let* $F$ be a left $A$-*module, $F', F''$ two submodules of $F$. Then: (i) The canonical mapping $B \otimes_A F' \to B \otimes_A F$ is an isomorphism of $B \otimes_A F'$ onto $BF'$. (ii) $F \cap BF' = F'$. (iii) $B(F' + F'') = BF' + BF''$. (iv) $B(F' \cap F'') = BF' \cap BF''$.

As $B$ is a flat right $A$-module, the canonical mapping

is injective; taking account of the identifications made, its image is $BF'$, which proves (i). Assertion (ii) follows from § 2, no. 6, Proposition 7, applied with $E = B$, $E' = A$, and using the formulae $A \otimes_A F = F$ and $A \otimes_A F' = F'$. Assertion (iii) is trivial and (iv) follows from § 2, no. 6, Proposition 6.