## Functorial Properties of the Tensor Product (Pt. I)

**Point of Post: **In this post we discuss some of the more “functorial” properties of the tensor product–namely that the tensor product defines a bifunctor, additive in each entry between certain categories.

*Motivation*

Now that we have defined tensor products we’d like to discuss some of the more category theoretic aspects of the construction. The tensor product functor shall serve as one of the prime examples of an additive functor.

*The Functorial Properties of the Tensor Product*

We begin by discussing the tensor product of two maps. Namely, suppose that are -bimodules and are -bimodules. Furthermore, suppose that we have an -map and an -map . We then want to define a map which is an -map such that . To prove the existence of such a map it suffices to prove that there exists an -biadditive -bimap such that . To do this we merely define . Let’s now show that is -biadditive as well as an -bimap. To see that is -biadditive we note that

to see that is a -bimap we merely note that

Thus, is an -biadditive -bimap and so ascends to an -map just that –moreover is unique to this property. We call this mapping the *tensor product *of the maps and and denote it by .

Now that we have figured out how to “tensor” two maps we can describe the tensor product as a functor:

**Theorem: ***Let be unital rings. Then, the tensor product is a covariant bifunctor , which is additive in each variable.*

**Proof: **The functor (just to be clear) takes to and takes to . To prove that this is, in fact, a functor it suffices to prove that and . Now, to prove these identities we note that since everything in sight is an -map it suffices to check these equalities on simple tensors. To do this we merely note that

and

By additive in each variable we mean that the functors and defined, as per usual with bifunctors, by and (and similarly for ) are additive functors. We prove this for since the other case is similar. So, we want to prove that induces a group homomorphism . Of course, this is equivalent to showing that if are -maps then . Once again, since both of these are -maps and so it suffices to check this on simple tensors. This is easy though since

so that .

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

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