Lemma 10.12.3. Let $M, N, P$ be $R$-modules, then the bilinear maps

\begin{align*} (x, y) & \mapsto y \otimes x\\ (x + y, z) & \mapsto x \otimes z + y \otimes z\\ (r, x) & \mapsto rx \end{align*}

induce unique isomorphisms

\begin{align*} M \otimes _ R N & \to N \otimes _ R M, \\ (M\oplus N)\otimes _ R P & \to (M \otimes _ R P)\oplus (N \otimes _ R P), \\ R \otimes _ R M & \to M \end{align*}

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