TITLE: Showing that $(X,Z) \cap (Y^2,X-YZ) \subseteq (XY,X-YZ)$ as ideals in $k[X,Y,Z]$.
QUESTION [0 upvotes]: I want to verify the following inclusion of ideals in $k[X,Y,Z]$:
$$(X,Z) \cap (Y^2,X-YZ) \subseteq (XY,X-YZ)$$
So far i had no success with any of my attempts. Still, i'd like to share what my latest attempt was.
My attempt:
$$a \in (X,Z) \cap (Y^2,X-YZ) \iff \begin{cases} a = f_1X+f_2Z \\ a = g_1Y^2+g_2(X-YZ) \end{cases}$$
Thus we consider \begin{align} f_1X+f_2Z = g_1Y^2+g_2(X-YZ) \\ f_1X+f_2Z = g_1Y^2+g_2X-g_2YZ \end{align}
What we want is showing, that $a$ can be presented as $$a = h_1XY + h_2(X-YZ)$$
I tried to tackle it with basic algebra, hoping i could somehow substitute and find suitable polynomials that do the trick, but i didn't succeed no matter what i've tried.
Could anyone help me showing how this is done? Please have some patience though, i'm still learning how to work with these kind of problems.
Thanks for any help!

REPLY [2 votes]: In your notation, $g_1Y^2\in(X,Z,X-YZ)=(X,Z)$, so $g_1\in(X,Z)$. Then $a\in(XY^2,Y^2Z,X-YZ)\subseteq(XY,X-YZ)$.
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