Maximal and Prime Ideal Consider a field $K$. I have seen that the ideal $(Y-X^2,Z-X^3)$ is a prime ideal in $K[X,Y,Z]$. But now I have to see that it is not a maximal ideal. Therefore, I have to find an ideal between $(Y-X^2,Z-X^3)$ and $K[X,Y,Z]$. Could $(X,Y-X^2,Z-X^3)$ be the maximal ideal I am looking for?

$(X,Y-X^2,Z-X^3)=(X,Y,Z)$, and thus the factor ring is isomorphic to $K$ as you were already told in the comments.
