# 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.