Corollary 1

If the left $A$-module $F$ admits a basis $(b_\mu)_{\mu \in M}$, every element of $E \otimes_A F$ can be written uniquely in the form $\sum_\mu (x_\mu \otimes b_\mu)$, where $x_\mu \in E$ and the family $(x_\mu)$ has finite support. The $\mathbf{Z}$-module $E \otimes_A F$ is isomorphic to $E^{(M)}$ considered as a $\mathbf{Z}$-module.

The basis $(b_\mu)$ defines an isomorphism of $F$ onto $\bigoplus_{\mu \in M} Ab_\mu$, whence there is an isomorphism $E \otimes_A F \to \bigoplus_{\mu \in M} (E \otimes_A Ab_\mu)$ by virtue of Proposition 7; as ξ ↦ ξb_μ is an isomorphism of A_s onto Ab_μ, x ↦ x ⊗ b_μ is an isomorphism of E onto E ⊗_A (Ab_μ) by virtue of Proposition 4 of no. 4, whence the corollary.

If E is a ((B'_i); A, (C'_j))-multimodule, the canonical isomorphism E ⊗_A F → E^{(M)} is a ((B'_i); (C'_j))-multimodule isomorphism.

In particular, if also E admits a basis (a_λ)_{λ ∈ L}, every z ∈ E ⊗_A F may be written in one and only one way in the form $\sum_{\lambda, \mu} (a_\lambda \xi_{\lambda\mu}) \otimes b_\mu$, where the ξ_{λμ} belong to A (and form a family of finite support); the mapping $z \mapsto (\xi_{\lambda\mu})_{(\lambda, \mu) \in L \times M}$ is an isomorphism of E ⊗_A F onto A^{(L × M)} for the \mathbf{Z}-module structures (and even the module structures over the centre of A). More particularly: