Better algebraic structures

Question

I normally like to think about algebraic structures as being built out of previously defined ones. So a ring is an abelian group with respect to addition, and a monoid with respect to multiplication, and the multiplication distributes over addition.

In a typical algebra class you do things like this until you get to fields, and we seem to stop there. Now I think this is because the nicest mathematical objects we come into contact with are fields, but I am wondering if there is anything "better". By that I mean satisfies more properties without adding an additional binary operation or other structure. If there are things like this, do people study them and why have I not heard of them?

In the same vein, is there anything better than a Hilbert Space?

Answer 1

There are structures which are "richer": for example, we can often have an action of a more complex structure upon a simpler one, so we can have groups acting on sets, and rings acting on abelian groups ($R$-modules, of which vector spaces are a particular kind: when the ring $R$ is a field).

It can also happen that we have a ring $A$ which is also an $R$-module (for some other commutative ring $R$), for which the action and ring multiplication are compatible, this ($A$) is an $R$-algebra (I'm being somewhat fast and loose with the definitions, here). If the "acting" ring is also a field, and the "acted on" ring is a division ring (which is almost a field), we get a division algebra, which is in many cases "more satisfying" than just a field. The canonical example here is the quaternions, which is both a division ring, and can be considered an algebra over the real number field.

The trouble here is, when we get "so special" with our structure, we find much less diversity in the examples. Nevertheless, the division algebra $\Bbb C$ (of dimension $2$ over the reals)is held in high esteem by many mathematicians for the "niceness" of its structure.