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R is a ring with 1.

We construct tensor product $M \otimes N$ of right R-module $M$ and left R-module $N$ to basically be able to state its universal property that any R-bilinear map from $M\times N$ to R-module P factors through an R-linear map from $M \otimes N$ to P. Atiyah and McDonald even mention that one can forget the construction if one wants to, we only need the universal property.

Given all this, can one do away with the process of constructing $M \otimes N$ and rather just prove the existence of one.

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marked as duplicate by Grigory M, user91500, user1729, Najib Idrissi, user88595 May 27 at 9:08

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Yes, the linked question gives a complete answer. I would vote to close if there weren't a bounty. –  Kevin Carlson May 25 at 22:18

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You can do it using category theory. Consider the functor $F:Ab\to Set$ given by sending $$A\to Bil_{R}(M,N,A)$$ where the notation on the right stands for bilinear maps into $A$. The tensor product is a representing object for $F$. That is $$Hom(M\otimes_{R}N,A)\simeq F(A)$$ where the bijection is natural. I suspect the existence of such an object follows from Brown representability, but I'll let you check the details for yourself.

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Brown representability is a statement about spaces. –  Qiaochu Yuan May 16 at 16:12
    
@Qiaochu Yuan Not true, there is a more general version which is applicable in any model category –  user43687 May 16 at 17:20
    
I also said, I suspect. I am not certain it will work. –  user43687 May 16 at 17:21

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