Proposition 6

Let E be a left A-module and F a right B-module. There exists one and only one $\mathbf{Z}$-homomorphism

mapping $y \otimes x$ to $y \otimes (1 \otimes x)$ for all $x \in E$ and all $y \in F$ and this homomorphism is bijective.

By definition the right hand side of (14) is $F \otimes_B (B \otimes_A E)$, where B is considered as a (B, A)-bimodule, and there is a canonical $\mathbf{Z}$-isomorphism $(F \otimes_B B) \otimes_A E \to F \otimes_B (B \otimes_A E)$ defined in § 3, no. 8, Proposition 8; on the other hand, the canonical isomorphism $F \to F \otimes_B B$ of § 3, no. 4, Proposition 4 is an isomorphism for the right A-module structures on the two sides, defined by $\rho$. Whence the desired isomorphism.

When A and B are commutative, the isomorphism (14) is an A-module isomorphism