Corollary 4

Let $M$ be a right $A$*-module, $N$ a left $A$*-module, $(x_i)_{1 \leq i \leq n}$ a family of elements of $M$, $(y_i)_{1 \leq i \leq n}$ a family of elements of $N$, such that $\sum_i (x_i \otimes y_i) = 0$ in $M \otimes_A N$. Then there exists a finitely generated submodule $M_1$ (resp. $N_1$) of $M$ (resp. $N$) containing the $x_i$ (resp. the $y_i$) and such that $\sum_i (x_i \otimes y_i) = 0$ in $M_1 \otimes_A N_1$.

$M$ (resp. $N$) is canonically identified with the direct limit of the right directed family of its finitely generated submodules containing the $x_i$ (resp. the $y_i$) and it suffices to apply Set Theory, III, § 7, no. 5, Lemma 1.