One of the things that has always driven my interest in Abstract Algebra as a field is the perceived multitude of *quirks*. It’s hard to speak of what I mean from a meta perspective, so I’ll just give an example.

Let be a field. Recall that a ring is called a -algebra if and if , where is the center of , that is, the elements for which for all . From an algebra perspective, any ring which is a -algebra is, necessarily, both a ring *and* a vector space over . For this reason, when speaking of being *generated* by a subset of elements of , it’s necessary to indicate with which regard the subset generates .

One cool example of this necessity – an example which I find *quirky*, in some regards – comes in the form of the polynomial ring in one variable over . Certainly with this construction, is a -algebra, and so is both a ring and a vector space over . Note, then, that as a ring, is a a finite-dimensional -algebra since is a ring generator for – that is, is the smallest ring over containing . On the other hand, has basis as a *vector space* over and hence is infinite dimensional as a -vector space.

Long story short: Two different structural existences for a single object, and the two are, to some extent, polar opposites as one another.

Things like this always make me realize why algebra is such a necessary and *beautiful* field.

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