Basic Properties of Tensor Products (Pt. II)
Point of Post: This is a continuation of this post.
Theorem(Associativity): Let be and bimodules respectively. Then, there is an isomorphism
of -bimodules such that
Proof: Fix and define
We claim that is an -biadditive -map. Indeed,
Thus, by the universal properties of tensor products we are afforded an -map with the property that . We now define a map
We claim that is -biadditive and -linear. Indeed,
To prove that we merely prove that for each fixed one has that the maps and are equal. Since both of these maps are -biadditive, it suffices to check that they agree on simple tensors. To do this we note that and . To prove that we once again, show that and agree on simple tensors, but this is trivial. Lastly, we want to show that but this merely requires, once again, checking on simple tensors, which is simple enough.
From all of this we see that lifts to an -map such that . Now, it’s clear that we could have repeated this same procedure, in the opposite direction, to produce a map
with the property that . It’s easy to see then that and are inverses, and so is an isomorphism.
Of course, by induction the above theorem allows us to freely associate any collection of tensor products and unambiguously denote a tensor product without parenthesising. Of course, the theoretical interpretation of what the tensor product of more than two modules means can be interpreted two-by-two, in the sense that can be seen as the module for which all -biadditive maps factor through. But, as we can see we can “further factor” this to see that we are really looking at modules which are universal for maps out of which are additive in each variable (with the others fixed) and which the appropriate scalars switch between adjacent entries. To be more explicit, let be modules respectively. Then, we call an -additive -bimap to be a map such that is additive in each variable, and such that
for all and . We see then , which in literal constructive terms is any associate of , is universal with respect to -additive -bimaps out of .
 Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
 Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.
 Blyth, T. S. Module Theory. Clarendon, 1990. Print.
 Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.
 Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.
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