We have defined a ring as a -module (abelian group) with a linear function satisfying certain properties. The concept of an algebra takes this definition and extends it to work over more general base rings than .
Let be a module over a commutative ring with unit. Then has both a left and a right action by , since is commutative. Thus, when we take the tensor product , the result is also an module. It makes sense, then, to talk about an -module homomorphism . Equivalently, this is a “multiplication” function such that
An -module equipped with such a multiplication is called an -algebra. We will often write the multiplication as . In many cases of interest, the base ring will be a field , but any ring is an algebra over .
Usually the term “algebra” on its own will refer to an associative algebra. This imposes an additional condition like the one we had in the definition of a ring: . An algebra may also have a unit so that for all . Algebras can also be commutative if for all elements . There are other kinds of algebras we’ll get to later that are not associative.
Pretty much everything I’ve said about rings works for associative algebras as well, substituting “-module” for “abelian group”. An -module is a left -module if there is an -linear function , and a similar definition works for right -modules. We can take direct sums and tensor products of -modules, and we have an -module of homomorphisms . All these constructions are clear from what we’ve said about modules over rings if we consider that is a ring, and that an -module is an abelian group with actions of both and which commute with each other.
The standard constructions of rings also work for algebras. In particular, we can start with an -module and build the free -algebra on like we built the free ring on an abelian group. Just use , where the tensor powers over make sense because is commutative.
We can also start with any semigroup and build the semigroup algebra just like we did for the semigroup ring . As a special case, we can take to be the free commutative monoid on generators and get the algebra of polynomials in variables over . In fact, almost all of “high school algebra” is really about studying the algebra , where is the field of rational numbers I’m almost ready to define.
Another source of -algebras extends the notion of the ring of endomorphisms. If is any -module, then is again an -module, and composition is -bilinear, making this into an -algebra.
Algebras over more general commutative rings than — particularly over fields — are extremely useful objects of study mostly because the linear substrate can often be much simpler. Building everything on abelian groups can get complicated because abelian groups can be complicated, but building everything on vector spaces over a field is generally pretty straightforward since vector spaces and their linear transformations are so simple.