## The Tensor Algebra and the Exterior Algebra (Pt. II)

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

Let us now compute an example.

Probably the simplest example is to compute when we think about as an abelian group. Indeed, we claim that, as -algebras (which is the same thing as rings), . Indeed, define by

Now, the important thing to note is that since we can move elements of in between the tensors that

and so it easily follows that is actually a ring map. That said, we know that and thus, clearly, as groups via the isomorphism that, on simple tensors, is just

So, it clearly suffices to find to find

But, following this map, it’s clear that is in the kernel if and only if and for . So, this just tells us that and for which tells us merely that is in . Thus, by the first ring isomorphism theorem we may conclude that as desired.

Note that this worked for , i.e. , as well, so that .

In fact, I don’t think it’s at all hard to see that (where the left tensor algebra is taken with respect to thinking of as the left regular -module).

**Graded Algebras and Graded Ideals**

We now discuss the notion of graded algebras, homgenous ideals ideals, graded quotients, and graded morphisms and how this applies to our study of the tensor algebra of a module.

Let be a ring. We say that is *graded *if there is distinguished decomposition (where each is a subgroup of ) with the property that (where by definition –i.e. the image of the multiplication map restricted to ). We call the *homogenous part of degree *and obviously call a *homogenous element of **degree .*

A *graded -algebra * is a graded ring such that the homogenous parts are -submodules.

The classic example of a graded -algebra is the polynomial ring with the elements of being the monomials of polynomial degree along with . This is clearly a graded ring for if and are monomials then or .

A *homogenous ideal *of a ring is an ideal such that with .

Now, if is a homogenous ideal of the ring then is naturally a graded ring with homogenous part of degree just being . I leave this to you to check. If is a graded -algebra then this ideal will also be a submodule and so the quotient will also be a graded -module with the already specified grading.

Now, suppose that and are graded -algebras. An -algebra map is called *graded *if it preserves homogeneous degrees–i.e. if .

For example, if is a graded -algebra and a graded ideal then the quotient map is a graded -algebra map.

Now, the reason we are discussing this in the tensor algebra post should be obvious–the tensor algebra is a graded -algebra. Indeed, the homogeneous part of degree in is just –these are all submodules and they satisfy .

**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.

[6] Conrad, Keith. “Exterior Powers.” *Www.math.uconn.edu/~kconrad*. Web. <http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/extmod.pdf>.

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