In this post, all rings are commutative with unit.
Let and be two rings. We say that is a -algebra, or an algebra over , if is also a -module, in such a way that the ring addition is the same as the module addition, and scalar multiplication satisfies
A morphism of -algebras is a -linear ring homomorphism. Explicitely:
- ;
- ;
- ; and
- .
The -algebras and their morphisms form a category, denoted .
Notice that given , we can produce . This gives us a quite “natural” function . This function is a ring homomorphism because it sends to and it respects addition by the module axioms, and it respects multiplication by the above axiom: On the other hand, given a ring homomorphism , we may use it to define scalar multiplication in terms of the multiplication in , and this gives a -algebra structure to . Hence the data of a -algebra structure on is equivalent, in a sense we’ll make precise, to a morphism of rings .
Fixing a ring , there’s the category of objects under , denoted by , which is a special case of the comma category construction where the objects are ring homomorphisms where ranges over the objects of , and where the arrows are ring homomorphisms such that the following diagram commutes:
Our claim is that there’s an isomorphism of categories (this is stronger than equivalence) between the category and the category of objects under . The proof is annoying so we omit it; constructing the correct functors that make up the isomorphism is quite easy. The only interesting part is: given an object in , we can define a -algebra structure on by specifying as the action of on , and given some -algebra , we can construct a ring homomorphism by sending to , just as we did earlier.
From this result, we can say that a -algebra structure on is precisely a ring homomorphism , which we call the structure morphism; now a -algebra homomorphism is a ring homomorphism that commutes with the structure morphisms.
Here is some more terminology. Let be an algebra.
We say is an algebra of finite type (French: algèbre de type fini) when there exists a finite set of elements of that are able to generate using the three available operations. In other words, an algebra is of finite type if and only if there is a surjective algebra homomorphism which sends each variable to a generator.
We say is a finite algebra if it is of finite type as a -module, that is, when there exists a finite set of elements of that are able to generate using only addition and scalar multiplication. Hence, an algebra is finite if and only if there is a surjective -module homomorphism sending the unit in each copy of to a generator.
An algebra over a field is finite if and only if is a finite-dimensional vector space over , which explains the terminology a little bit.