Point of Post: In this post we discuss the notion of -algebras where is some commutative unital ring, and the associated categories.
Very often when we run into -modules (where we assume that is commutative and unital) there is more structure involved, namely the modules are also rings which interact nicely with the -module structure. Namely, we have a (left, right)-module along with a bilinear map which makes into a unital ring such that . We have already run into algebras before, in the context of endomorphism algebras of vector spaces. More generally, any ring of matrices is given the structure of an -algebra. In fact, it’s easy to see that algebras generalize ring theory since, as we shall see, the category of all -algebras is isomorphic to the category of rings. Other examples of algebras are polynomial algebras and is a -dimensional -algebra. Algebras play an important role in a lot of algebraic subjects, perhaps most notably with their appearance in differential geometry in the form of tensor algebras, and their appearance in commutative algebra.
Let be a commutative unital ring, then we call an -module (no need for distinction about left or right since is commutative) together with a -bilinear map such that is a unital ring, and for all to be an -algebra.
There is another common definition of an-algebra. Namely, let be a unital ring and let be a unital ring homomorphism, we’d like to consider this the same thing as an -algebra.
Indeed, we can define an -module structure on by defining . To see this, we merely note that
Moreover, this module structure along with the usual ring structure of makes into an -algebra. The only thing to check is that but this follows since this can be rewritten as which follows from the associativity of and the fact that . Moreover, the fact that is unital follows from the fact that for all from where the conclusion follows.
Conversely, if is an -algebra we can define a map by defining . To see that this is really a unital -map we note that
The fact that follows from the fact that for all and .
To consider some examples, we note that for any commutative unital ring the polynomial rings and the matrix rings . In fact, more generally, given any -algebra we have that and are -algebras with the usual notions of addition and multiplication.
If is a topological space then the set of all continuous maps is an -algebra, similarly the set of all continuous maps is a -algebra.
Any ring is a -algebra in the natural way whenever .
The Hamiltonian quaternions form a -dimensional -algebra.
Let be some field and a (not necessarily finitely dimensional) -space. Then, the endomorphism algebra is an algebra over .
We define a map of -algebras or -algebra homomorphism from the -algebra to the -algebra to be a set function which is both a unital ring homomorphism and an -map. If we think about the -algebras and as just rings and with unital ring homomorphism and . Then, an -algebra map is nothing more than a unital ring map such that .Indeed, one merely checks that implies that
so that is an -map as well.
As an example of an (iso)morphism of -algebras one can recall the basic fact from linear algebra that if is a field and is an -dimensional -space then .
We then define to be the category of all -algebras with -algebra homomorphisms. We define to be the full subcategory whose objects are -algebras such that is commutative. It’s fairly easy to see that there is only one way to define a -algebra structure on a given ring , this is because is initial in and so there is only one unital homomorphism . Moreover, it’s easy to verify that this is just the usual -module structure on the underlying abelian group of . Thus, we see that and .
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 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.