ag.algebraic geometry - Open affine subscheme of affine scheme which is not principal

For a simple, really concrete example you can also look at:

$A=k[x,y,u,v]/(xy+ux^2+vy^2)$, $X =Spec(A)$, $I=(x,y)$, $U = D(I)$.

Then the functions $f=frac{-v}{x}=frac{y+ux}{y^2}$ and $g=frac{-u}{y}=frac{x+vy}{x^2}$ are defined on $U$. But $yf+xg=1$, so $U$ is affine!

Cheers,