Let K be a field, and $I=(XY,(X-Y)Z)⊆K[X,Y,Z]$. Prove that $√I=(XY,XZ,YZ)$.

Question

Let $K$ be a field, and let $I=(XY,(X-Y)Z) \subset K[X,Y,Z]$. Prove that $\sqrt{I}=(XY,XZ,YZ)$.

I have no idea how to start with this question, can anybody give me some hint?

Thanks a lot.

Answer 1

I will to use the following fact

Let $A$, $B$, $C$ be elements of $ \mathbb{k}[x_1,\dots x_n]$. We have the following identity between radical ideals: $$\sqrt{(A,BC)}=\sqrt{(A,B)}\cap\sqrt{(A,C)}$$

In our case, using this useful fact, we obtain the decomposition $$\sqrt{(XY,(X-Y)Z)}=\sqrt{(X,(X-Y)Z)}\cap \sqrt{(Y,(X-Y)Z)}= \sqrt{(X,YZ)}\cap \sqrt{(Y,XZ)}$$ In the same way we have $$\sqrt{(Y,XZ)}=\sqrt{(Y,Z)}\cap \sqrt{(Y,X)}$$ and $$\sqrt{(X,YZ)}=\sqrt{(X,Z)}\cap \sqrt{(X,Y)}$$ Using this identities we obtain $$\sqrt{(XY,(X-Y)Z)}=\sqrt{(X,Z)}\cap \sqrt{(X,Y)} \cap \sqrt{(Y,Z)}=(X,Y)\cap (X,Z)\cap(Y,Z)$$ as required.