Corollary 2

If E and F are two free modules over a commutative ring C and (aλ) (resp. (b_μ)) is a basis of the C-module E (resp. F), then (a_λ ⊗ b_μ) is a basis of the C-module E ⊗_C F.*

By an abuse of language, the basis (a_λ ⊗ b_μ) is sometimes called the tensor product of the bases (a_λ) and (b_μ).

Remark (1). Let E be a free right A-module, F a free left A-module, (a_λ)_{λ ∈ L} a basis of E and (b_μ)_{μ ∈ M} a basis of F. Every element z ∈ E ⊗_A F may be written uniquely as $\sum_\lambda a_\lambda \otimes y_\lambda$, where y_λ ∈ F, and also uniquely as $\sum_\mu x_\mu \otimes b_\mu$, where x_μ ∈ E. If we write $y_\lambda = \sum_\mu \eta_{\lambda\mu} b_\mu$, $x_\mu = \sum_\lambda a_\lambda \xi_{\lambda\mu}$, where the ξ_{λμ} and η_{λμ} belong to A, then $\xi_{\lambda\mu} = \eta_{\lambda\mu}$ for all (λ, μ), for