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I am reading through this book and the following proof for the distributivity of the tensor product has me confused.

Let $A$ be a ring and let $M_i,N$ be $A$-modules. Then we have a canonical isomorphism $$ \left( \oplus_{i \in I} M_i \right) \otimes_A N \simeq \oplus_{i \in I} \left( M_i \otimes_A N \right) $$

The proof:

Let $\phi : \left( \oplus_{i \in I} M_i \right) \times N \to \oplus_{i \in I} \left( M_i \otimes_A N \right)$ be the map defined by $\phi : \left( \sum_i x_i,y\right) \mapsto \sum_i \left( x_i \otimes y \right)$.

Let $f : \left( \oplus_{i \in I} M_i \right) \times N \to L$ be a bilinear map. For every $i \in I$, $f$ induces a bilinear map $f_i : M_i \times N \to L$ which factors through $\tilde f_i : M_i \otimes_A N \to L$. One verifies that $f$ factors uniquely as $f = \tilde f \circ \psi$ with $\psi : \left( \oplus_{i \in I} M_i \right) \times N \to \left( \oplus_{i \in I} M_i \right) \otimes_A N$ the canonical map and $\tilde f = \oplus_i \tilde f_i$.

Hence $\oplus_{i \in I} \left( M_i \otimes_A N \right)$ is the tensor product of $\left( \oplus_{i \in I} M_i \right)$ with $N$.

I am confused since the map $\phi$ defined at the start of the proof doesn't occur anywhere else. I also don't see how the Hence follows.

Please note I have seen a couple proofs of the statement before and am convinced of its validity. I am only interested in understanding Qing Liu's proof.

Can anyone help?

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I think it's a mistake. Liu's sentence

One verifies that $f$ factors uniquely as $f = \tilde f \circ \psi$ with $\psi : \left( \oplus_{i \in I} M_i \right) \times N \to \left( \oplus_{i \in I} M_i \right) \otimes_A N$ the canonical map and $\tilde f = \oplus_i \tilde f_i$.

should be replaced by:

One verifies that $f$ factors uniquely as $f = \tilde f \circ \phi$ with $\tilde f = \oplus_i \tilde f_i$. This shows that the pair $\left( \bigoplus_{i\in I} \left( M_i \otimes N \right) , \phi \right)$ satisfies the universal property of the tensor product of $\bigoplus_{i \in I} M_i$ and $N$. But there is only one pair (up to isomorphism) satisfying this property.

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    $\begingroup$ Thanks. This is what I also suspected. It also makes sense since the $\tilde f_i$ come from $M_i \otimes N$, so the $\oplus \tilde f_i$ should come from $\oplus (M_i \otimes N)$. In Qing Liu, his $\psi$ doesn't seem to map here, so the whole thing becomes ugly. $\endgroup$
    – Ruben
    Sep 27 '19 at 16:41
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    $\begingroup$ I suspect Liu had two different proofs in mind, and ended up with a hodgepodge of the two. $\endgroup$ Sep 27 '19 at 16:42

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