Why are "algebras" called algebras?
There's a mathematical object called an "algebra" (e.g. an algebra over a ring), but why does this particular object have such an "important" name (which makes it sound like the most important concept in this huge area, abstract algebra), whereas the names of other important algebraic structures such as magmas, groups, rings, lattices and modules sound less important. I know some universal algebra and category theory, so I understand that "algebras" have many kin objects. But I can't understand why somebody decided to call these particular objects "algebras", although there seem to be many other good candidates for this grand name.
Similarly, there are objects called "numbers" in number theory, "sets" in set theory, "categories" in category theory, and "topologies" in topology. However, in other areas, for example analysis, geometry and even mathematics, there is no object called an "analysis", a "geometry" or a "mathematics". Is it because there's no fundamental object in these areas which have unsurpassable importance over others? If there are any central objects in these areas which should be named by their significance, in the same way as the objects "category", "topology", and "set" in their respective areas, could you tell me them?
Well of course this has historic reasons. I don't know the details, though. But I would like to explain why the notion of an algebra over a ring, suitably generalized, is fundamental.
There are various notions which look very similar:
- ring
- monoid
- algebra over a ring
- normed algebra
- Banach algebra
- sheaf of rings
- topological monoid
- topological ring
- ring spectrum
- ...
Category theory is the unique field of mathematics where "similar" things are united to "one" thing. And in fact, in the context of monoidal categories, the mentioned examples are actually instances of one single notion: Monoid object, often also called "algebra object". One just has to apply this notion to different monoidal categories. In the above examples, these are:
- abelian groups
- sets
- modules over a ring
- normed vector spaces
- Banach spaces
- sheaves of abelian groups
- topological spaces
- topological abelian groups
- symmetric spectra