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Let $\otimes_{i \in I} M_i$ denote the tensor product of $R$-modules $M_i$. Then $\otimes_{i \in \emptyset} M_i$ is $R$. (Reason: $\prod_{i \in \emptyset} M_i$ is the terminal object in the category of sets (see the answer of Eivind Dahl), i.e. a point, and multilinear maps on this to $N$ are just elements of $N$, i.e. homomorphisms $R \to N$.) Another one: I know this is really silly and already contained somehow in the other answers, but anyway: $\prod_{i \in \emptyset} 0 = 1$ |
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Let $\otimes_{i \in I} M_i$ denote the tensor product of $R$-modules $M_i$. Then $\otimes_{i \in \emptyset} M_i$ is $R$. (Reason: $\prod_{i \in \emptyset} M_i$ is the terminal object in the category of sets (see the answer of Eivind Dahl), i.e. a point, and multilinear maps on this to $N$ are just elements of $N$, i.e. homomorphisms $R \to N$.) |
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Let $\otimes_{i \in I} M_i$ denote the tensor product of $R$-modules $M_i$. Then $\otimes_{i \in \emptyset} M_i$ is $R$. |
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