i have some problems with the calculations in the ideals. I recently studied algebraic geometry and the concept of projective closure. There is a counterexample which has just homogenized the base of an ideal, does not give the projective closure of the twisted cube. You will find it under: Projective closure of the affine curve

I don't see why the polynomial $ p_1 = y ^ 2-xz $ is an element of the ideal $ (x ^ 2-y, x ^ 3-z) $. I know that the ideal is generated in a finite way. So I have to solve the following equation $ p_1 = r_1 * (x ^ 2-y) + r_2 * (x ^ 3-z) $ for $ r_1, r_2 in k (x, y, z) $, which I find quite difficult. Are there easy arithmetic rules for ideals?

This brings me to a more general reflection. For example: $ frac {k (x, y, z)} {(z-1, x ^ 2-y)} = k (x, x ^ 2) $ there you use it $ z = $ 1 and $ y = x ^ 2 $. In the same sense, you can show that $ (x ^ 3-z, x ^ 2-y, xy-z) $ is generated by only two elements:

$ (- 1) (x ^ 2-y) * x = (x ^ 3-y * x) (- 1) = (z-xy) (- 1) $

last equation is valid because $ x ^ 3 = z $. (I saw it in a conference)

Can i use it for problem? If this is true, why is it mathematically correct?