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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.

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    The quotient of a polynomial ring by a single polynomial is already a pretty nice ring. I'm not sure what kind of answer you're expecting. What do you actually want to know?2011-05-10

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