Is $\Bbb{Q}[x,y]/(x)=\Bbb{Q}[y]$? Is $\Bbb{Q}[x,y]/(x)=\Bbb{Q}[y]$ for all practical purposes?
I used to think so, but my friend says that there are subtle differences between the two. I fail to grasp them. 
 A: $\mathbb{Q}[x, y]/(x)$ and $\mathbb{Q}[y]$ are isomorphic (and via a very nice, natural isomorphism with many good properties). But whether the two rings are equal depends on the precise details of how they are represented as mathematical objects. The standard formal representation of a polynomial in $x$ and $y$ over a field $F$ is as a finite partial function mapping monomials $x^my^n$ to non-zero elements of $F$. E.g., $x^2 + 2xy + y^2$ is represented by the function given by the set of pairs $\{(x^2, 1), (xy, 2), (y^2, 1)\}$. (Here $x^my^n$ stands for some formal representation of the monomial, e.g., as the pair $(m, n)$.)
With this representation, you do have some equalities, e.g., $F[x]$ is equal to a subring of $F[x, y]$. However, the quotient ring $F[x, y]/(x)$ (as usually defined) comprises equivalence classes of a relation on $F[x, y]$ and so its elements are sets of finite partial functions and are not themselves finite partial functions. So $F[x, y]/(x) = F[y]$ is not true.
You could come up with a contrived system of representations that did make $F[x, y]/(x)$ and $F[y]$ equal, but I don't know of any natural way of doing it and would be rather surprised if there was one.
