Let $A,B,C$ be R-modules. I know that _$\otimes C$ is an additive functor so $(A \otimes C) \oplus (B \otimes C) \cong (A \oplus B) \otimes C$ but what should the isomorphism send $(a \otimes c,b \otimes \hat{c})$ to?
2026-04-01 10:27:54.1775039274
$(A \otimes C) \oplus (B \otimes C) \cong (A \oplus B) \otimes C$
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Why not $a \otimes c + b \otimes \hat{c}$. Nobody says the image has to be a pure tensor.