I can't find the mistake in my logic and so I hope you can help me.
Let $k$ be an algebraically closed field. It is well known that the projective algebraic sets $\mathbb{P}_{k}^1$ and $V=V(x^2+y^2-z^2)$ are isomorphic. In an exercise problem, we were supposed to prove that coordinate rings of rational curves (i.e. curves which are birationally equivalent to $\mathbb{P}^1$) are UFDs. However, $k[V] = k[x,y,z]/(x^2+y^2-z^2)$ is not a UFD (see e.g. MSE/413506
).
Where is my mistake?
Best Answer
I think you need to take the affine coordinate ring of the curve not the projective coordinate $k[x, y, z]/(x^2+y^2-z^2)$ . So in this example, the question is is $k[x,y]/(x^2+y^2-1)$ a UFD, and I think so since $k$ is algebraically closed:
Ring of trigonometric functions with real coefficients