What goes wrong in the proof that $Spec \mathbb{R}[x,y]/(x^2+y^2)$ is a principal divisor in $Proj \mathbb{R}[x,y,z]/(x^2+y^2+z^2)$

Question

Let $C=\text{Proj}\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$. Consider a closed subset $Z=\{z\neq 0\}$.

I claim that $Z=(f)$ with $f=x^2+y^2\in K(C)$.

Indeed, if $z=0$ then the we get exactly $Z$. If $z\neq 0$, then $f$ has no poles because $\frac1f=\frac1{x^2+y^2}=-1$ has no zeros, and $f$ has no zeros at $z\neq0$ because we get $C=\text{Spec}\mathbb{R}[x,y]/(x^2+y^2+1)$ and forcing $x^2+y^2=0$ we get $C=\text{Spec}\mathbb{R}[x,y]/(1)=\emptyset$.

I know the claim is false because on smooth curves principal divisors have degree zero, while $Z$ above is effective. What is wrong with my argumenation?