Radical of ideal 
I am trying to compute the radical of ideal $I=((X-Z)(X-Y)(X-2Z),X^2-Y^2Z)$. 

I suspect that rad$(I)= (X-Z) \cap (X-Y) \cap (X-2Z) \cap (X^2-Y^2Z)$ which is $(X-Z)(X-Y)(X-2Z)(X^2-Y^2Z)$. Is my calculation right? If not, then what is radical of the ideal and how to compute it?
 A: I thought about the problem for a while but the easiest effective approach I could think of was brute-force application of Nullstellensatz.
Let me call your ideal $\mathfrak a$ to avoid confusion with the set of vanishing polynomials:
Calculate vanishing set $V(\mathfrak a)$ of $\mathfrak a$: let $(x,y,z)\in \Bbb C^3$ such that all elements of $\mathfrak a$ vanish for $(x,y,z)$. Then there are three cases:


*

*$x=y$ and ($z=1$ or $x=0$).

*$x=z$ and ($y^2 =x$ or $x=0$).   

*$x=2z$ and ($y^2=2x$ or $x=0$).


So $V(\mathfrak a)$ is the set of triples fulfilling 1,2 or 3, so it is the union of the three sets $V_1,V_2,V_3$ fulfilling 1,2 or 3 respectively.
So to compute $r(\mathfrak a) = I(V(\mathfrak a)) = I(V_1 \cup V_2 \cup V_3)=I(V_1) \cap I(V_2) \cap I(V_3)$ we need the  polynomials vanishing for $V_1$, $V_2$ and $V_3$ respectively. These are:
$\\ I(V_1) = (X-Y,(Z-1)X) \\ I(V_2) = (X-Z,(Y^2-X)X)  \\ I(V_3) = (X-2Z,(Y^2-2X)X)$
So $r(\mathfrak a) = (X-Y,(Z-1)X) \cap (X-Z,(Y^2-X)X)  \cap (X-2Z,(Y^2-2X)X)$.
