Proposition 4

If $E$ is a right $A$-module, the mapping $h : x \mapsto x \otimes 1$ of $E$ into $E \otimes_A (sA_d)$ is a right $A$-module isomorphism, whose inverse isomorphism $g$ is such that $g(x \otimes \lambda) = x\lambda$ for $x \in E, \lambda \in A$.

If $g$ is the canonical mapping, $g \circ h$ is the identity mapping $1_E$ and $h \circ g$ coincides with the identity mapping of $E \otimes_A (sA_d)$ onto itself for elements of the form $x \otimes y$, which generate the latter $\mathbf{Z}$-module; hence the conclusion.

We shall normally write $E \otimes_A A$ instead of $E \otimes_A (sA_d)$ and shall often identify $E \otimes_A A$ with $E$ by means of the above canonical isomorphisms. Observe that, if $E$ also has a (left or right) $B$-module structure which is compatible with its right $A$-module structure, $g$ and $h$ are also isomorphisms for the $B$-module structures on $E$ and $E \otimes_A A$ (and hence multinode module isomorphisms).

Now let $F$ be a left $A$-module; $(sA_d) \otimes_A F$ (also denoted by $A \otimes_A F$) then has a canonical left $A$-module structure and as in Proposition 4 a canonical isomorphism is defined of $A \otimes_A F$ onto $F$ mapping $\lambda \otimes x$ to $\lambda x$, and its inverse isomorphism is $x \mapsto 1 \otimes x$.

In particular there exists a canonical isomorphism of the $(A, A)$-bimodule $(sA_d) \otimes_A (sA_d)$ onto $sA_d$ which maps $\lambda \otimes \mu$ to $\lambda \mu$.