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    Commutative rings

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    Show that if A and B are commutative rings and if f : A → B is a ring homorphism, then there is a
    natural B-module isomorphism from:
    (B ⊗A M) ⊗B (B ⊗A N)
    B ⊗A (M ⊗A N)
    for A-modules M,N

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    Let Mod_A be the category of A-modules, and Mod_B be the category of B-modules. We have the functor F: Mod_AMod_B, sending an A-module M to the B-module B⊗_AM with b(b'⊗x)=bb'⊗x, for b,b'∈B and x∈M. The functor F is called the extension of scalars functor. It is easy to note that the first expression in the problem is F(M) ⊗_A F(N), while the second one is F(M⊗_A N) . Therefore, we need to ...

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    This solution helps with a question about communicative rings.