Is there a better way of writing this ring? The quotient $k[x,y]/(x-y^2)$ is isomorphic to $k[y]$  as a ring. Suppose, $g$ is a polynomial in $y^2$. Is there a "nice" ring that is isomorphic to $k[x,y^2]/(x^2-gy^2)$ assuming $g$ is not a unit?
Edit: Sorry I meant to write a different second ring.
 A: I assume you mean  $k[x,y]/(x^2-gy^2)$ . I think you won't be able to find a "nice ring" isomorphic to your ring because geometrically you have an affine curve of positive genus (except in some degenerate cases e.g. $g=y$) .These curves are not rational and so not isomorphic to an open subset of $\mathbb A^1_k$. In particular the  ring of these curves can't be something as simple as, say, $k[x]$ or $k[x,x^{-1}]$.[We have to be careful with the notion of genus since the curve is singular and not complete, but this is technical and not really crucial ]
Conclusion I'm not sure how happy you'll be with this answer, since "niceness" is in the eye of the beholder. Let me sum up by saying that geometers have come up with an invariant for an algebraic curve, its geometric genus, which is an integer that gives a rough indication of the complexity of that curve, from an algebraic, geometric and arithmetic point of view.
A: The only "nice" way I can think of to write that would be $k[y,y\sqrt{g}]$. I don't think it can be simplified any further unless $g$ is a square in $k[y]$.
Edit: My answer applies to a previous version of the question.
