The Stacks project

Lemma 10.12.4. Let $M_1, \ldots , M_ r$ be $R$-modules. Then there exists a pair $(T, g)$ consisting of an $R$-module T and an $R$-multilinear mapping $g : M_1\times \ldots \times M_ r \to T$ with the universal property: For any $R$-multilinear mapping $f : M_1\times \ldots \times M_ r \to P$ there exists a unique $R$-module homomorphism $f' : T \to P$ such that $f'\circ g = f$. Such a module $T$ is unique up to unique isomorphism. We denote it $M_1\otimes _ R \ldots \otimes _ R M_ r$ and we denote the universal multilinear map $(m_1, \ldots , m_ r) \mapsto m_1 \otimes \ldots \otimes m_ r$.