Abstract Algebra Quirks

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 k be a field. Recall that a ring R is called a k-algebra if k\subset Z(R) and if 1_k=1_R, where Z(R) is the center of R, that is, the elements x\in R for which xy=yx for all y\in R. From an algebra perspective, any ring R which is a k-algebra is, necessarily, both a ring and a vector space over k. For this reason, when speaking of R being generated by a subset of elements of R, it’s necessary to indicate with which regard the subset generates R.

One cool example of this necessity – an example which I find quirky, in some regards – comes in the form of the polynomial ring R=k[x] in one variable over k. Certainly with this construction, R is a k-algebra, and so R is both a ring and a vector space over k. Note, then, that as a ring, R is a a finite-dimensional k-algebra since x is a ring generator for R – that is, R=k[x] is the smallest ring over k containing x. On the other hand, k[x] has basis 1,x,x^2,\ldots as a vector space over k and hence is infinite dimensional as a k-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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