Corollary 2

Let $E^0$ (resp. $F^0$) denote the module $E$ (resp. $F$) considered as a left (resp. right) module over the opposite ring $A^0$; then there exists one and only one $\mathbf{Z}$*-module isomorphism* $\sigma : E \otimes_A F \to F^0 \otimes_{A^0} E^0$ such that $\sigma(x \otimes y) = y \otimes x$ for $x \in E$ and $y \in F$ ("commutativity" of tensor products).

By definition of the $A^0$-module structures on $E^0$ and $F^0$, the mapping $(x, y) \mapsto y \otimes x$ of $E \times F$ into $F^0 \otimes_{A^0} E^0$ is $\mathbf{Z}$-bilinear and satisfies conditions (1), whence the existence and uniqueness of the $\mathbf{Z}$-linear mapping $\sigma$. Similarly a $\mathbf{Z}$-linear mapping $\tau : F^0 \otimes_{A^0} E^0 \to E \otimes_A F$ is defined such that $\tau(y \otimes x) = x \otimes y$ and clearly $\sigma$ and $\tau$ are inverse isomorphisms.