Proposition 5

Let E be a left A-module and F a left B-module; the composite mappings

are respectively equal to the identity mappings of $\rho^*(E)$ and $\rho_*(F)$.

We give the proof, for example, for (12); for all $x \in E$, the mapping $\rho^*(\phi_E)$ maps $1 \otimes x$ to the element $1 \otimes (1 \otimes x)$ and the mapping $\psi_{\rho^*(E)}$ maps $1 \otimes (1 \otimes x)$ to the element $1 \otimes x$; the conclusion follows from the fact that the elements of the form $1 \otimes x$ generate the B-module $\rho^*(E)$; the proof is even simpler for (13).