/sci/ - Science & Math

The coordinate ring of the parabola is [math]k[x][/math] and of the hyperbola is [math]k[x,x^{-1}][/math] by simple computation. Let's assume for simplicity that we're looking at a unit circle, not ellipse. It's coordinate ring is [math]k[x,y]/(x^2+y^2-1)[/math] so every element can be written in the form [math]ay+f(x)[/math] with [math]f(x)\in k[x][/math] and [math]a\in k[/math], since the higher powers of [math]y[/math] get reduced to linear terms. How can I show that this ring is isomorphic to either of the other two?

It looks enticingly like [math]k[x][/math] but it is not a UFD...