Corollary 5

Let E be a right A-module, F a left A-module, M a submodule of E and N a submodule of F. If M is a direct factor of E and N a direct factor of F, the canonical homomorphism $M \otimes_A N \to E \otimes_A F$ is injective and the image of $M \otimes_A N$ under this homomorphism is a direct factor of the $\mathbf{Z}$-module $E \otimes_A F$.

This follows immediately from Proposition 7.

Note that if E is a $((B'_i); A, (C'_j))$-multimodule and F an $(A, (B''_h); (C''_k))$-multimodule and M and N direct factors in these multimodules, $M \otimes N$ is a direct factor of the $((B'_i), (B''_h); (C'_j), (C''_k))$-multimodule $E \otimes F$.