I hope I'm not overbearing in this site. Yes, I'm still struggling.
If you can, I have a question about primary decomposition that still needs help, you can find it in my page.
Now I wanted to find the height of a certain ideal. In the ring $ A = K[x,y,z]/(xy,xz,z-y^2)$ I need to find the height of $I = (x,y,z)$. Here is my reasoning up to now:
the Krull theorem states that the height of a prime ideal, such as $I$, is not greater than the number of generators of $I$, so it's not greater than the least amount of polynomials that generate $I$. Since in $A$, $z=y^2$, we can consider $I$ having 2 generators, so the height of $I$ is at best two. So I'd need to find two proper prime ideals that are contained in $I = (x,y)$. I really don't know how to use the conditions $xy=0$ and $xz=0$, so I'm having second thoughts about my solution. I initially said that the height is indeed two because $(0) \subseteq (x) \subseteq I$ and I don't need to find if there are any more ideals there, but I just realized $(0)$ is not prime because the ring is not a domain since $x*y=0$, but $x,y \neq 0$. Can somebody help me? Thank you!
