My teacher give us an exercise to determine if the ideal of a polynomial ring k[x,y,z](where k is an algebraically closed field) can be decomposed into smaller ideal. The ideal is $I$=($x^{2}-yz$,$xz-x$). The answer is that $I$=($y,x$)$\cap$($z,x$)$\cap$($x^{2}-y,z-1$).
I know that $I$ is the intersection of $(x^{2}-yz,x)$ and $(x^{2}-yz,z-1)$, but then I thought it cannot be further decompose. How to arrive the given answer?
The first ideal is the same as $(yz, x)$. Thinking about the corresponding geometric set of points (scheme, variety, whatever), this is $x=0$ and $yz=0$. That is, the union of two lines in the $x=0$ plane.