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
See the attachment.
Let Mod_A be the category of A-modules, and Mod_B be the category of B-modules. We have the functor F: Mod_AMod_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 ...
This solution helps with a question about communicative rings.