## The Construction of the Tensor Product of Modules (Pt. III)

**Point of Post: **This is a continuation of this post.

Once again, this concretely means that if is an -bimodule and a -bimodule then an tensor product of by over is an module and an -biadditive -map which has the property such that given any other -bimodule and an -biadditive -map there exists a unique -map such that .

For example, the previous construction of a tensor product as an abelian group corresponded to -tensor products over .

Once again, we have by mere definition the following important theorem:

**Theorem: ***Let be a -bimodule and a -bimodule. Then, if and are two -tensor products over , of by then there exists a unique -isomorphism such that .*

Unfortunately, similar to before there is absolutely no indication that -tensor products over a ring should exist in general. Strangely enough, we have the following theorem that brings together all of the theory:

**Theorem: ***Let be an -bimodule and an -bimodule. Then, the multiplications and extend to give a -bimodule structure on , and moreover is then a -tensor product of by over .*

**Proof: **We begin by showing that there really is a (necessarily unique) -multiplication on for which . Indeed, consider the map given by . We note then that for any , and we have that

so that is -biadditive. Thus, by the universal property of tensor products we get a group homomorphism such that . We then define the multiplication map by and denote, as per usual, as just . We note though that since

and maps on are determined by their values on elements of the form we may conclude that . Similarly, one can check that on elements of the form and and so they are equal on all of . We then proceed to verify the desired axioms

Thus, we see that we have defined a left -multiplication on such that as desired. The exact same process shows that we have a right -multiplication on such that . To prove that everything defines a -bimodule structure on it suffices to show that for all , , and . It of course suffices to do this on simple tensors:

thus this really is a -bimodule structure on .

Let’s now verify that really is an -tensor product of by over . To do this let’s first check that is an -biadditive -map. The fact that it’s -biadditive follows from previous note, and so it suffices to show that it is -biadditive. To do this we merely note that

and so the fact that it’s an -map follows.

Let’s now show that it’s initial in the desired category. To this end suppose that we have some -bimodule and an -biadditive -map . Since is -biadditive we are guaranteed, by construction, a group homomorphism such that , it thus remains to show that is an -map. To do this let be arbitrary and we see then that

Thus, is an -map such that . Since the uniqueness is guaranteed by the construction of the fact that is an -tensor product over follows.

**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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