I have the following past exam paper question, a similar sort of question seems to come up every year.. And I'm completely lost with it...

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $(y^2-xy-2zx, y^3+z^2+1, x^2yz-zy)$.

Show that $J$ is zero-dimensional.

What is the dimension, as a vector space over $\mathbb{Q}$, of $\mathbb{Q}[x,y,z]/J$?

Define the ideal $I := (xy^2+2xz-yz, x^2yz+y^2)$. Explain why $I$ is not zero-dimensional.

I know the definition for an ideal to be zero-dimensional, but I cant find an example of how to use to it to show an ideal is zero-dimensional...

Defintion. An ideal $I\subset K[X_1,...,X_n]$ is *zero-dimensional* if the $K$-vector space $K[X_1,...,X_n]/I$ is finite dimensional.