Lemma 3

For a tight $A$-module $E$ to be $M$-flat, it is necessary and sufficient that for every finitely generated submodule $M'$ of $M$ the canonical homomorphism

(*j being the canonical injection $M' \to M$) be injective.

Suppose that this condition holds and let $N$ be any submodule of $M$. Suppose that the canonical image in $E \otimes_A M$ of an element

is zero and let $M'$ be the finitely generated submodule of $N$ generated by they.; as by hypothesis the composite mapping $E \otimes_A M' \to E \otimes_A N \to E \otimes_A M$ is injective, the sum $z' = \sum x_i \otimes y_i$, considered as an element of $E \otimes_A M'$, is zero. As $z$ is the image of $z'$, we have also $z = 0$, whence the lemma.